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<_)-k I' 



THEORY AND APPLICATIONS 



OF 



FINITE GROUPS 



BY 

G. A. MILLER 

rROFBSSOR OF MATHEMATICS IN THE UNIVERSITY OF ILLINOIS 

H. F. BLICHFELDT 

PROFESSOR OF MATHEMATICS IN STANFORD UNIVERSITY 

L. E. DICKSON 

PROFKSSOK OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO 



FIRST EDITION 

FIRST THOUSAND 



I 



NEW YORK ^ ^ > 

JOHN WILEY & SONS, Inc. \\y\/ V 

London: CHAPMAN & HALL, Limited ^y\{0 ^ 
1916 (r'l 



Copyright, 1916^ 

BV 

G. A. MILLER, H. F. ULICHFKLDT and L. E. DICKSON 



m 



THK SCIENTIFIC FRESS 

KOMKItT ORUMMONO ANO COMPANY 

BROOKLYN. N. Y. 



To 
Catnille Jorban, 



whose fundamental investigations 

on the theory and .vpplications of finite groups 

enriched the subject to the extent of converting it 

into a fundamental branch of mathematics 

and furnished in a large measure the 

inspiration for the subsequent 

great activity in this field, 

this book is dedicated 

(by permission) 



PREFACE 



The aim of this book is to present in a unified manner the 
more fundamental aspects of finite groups and their applications, 
and at the same time to preserve the advantage which arises 
when each branch of an extensive subject is written by one who 
has long specialized in that branch. 

To secure unification, the three authors planned the book 
after various conferences and extensive correspondence, while 
each read and commented upon both the MS. and proof-sheets 
of the parts by the remaining authors. However, the influence 
of each author upon the other two has been mainly of editorial 
character, so that the individuality of the authorship of each 
part remains intact. 

Part I, written by G. A. Miller, gives in the first two chapters 
various concrete examples of groups and an elementary presen- 
tation of the most fundamental theorems on groups of sub- 
stitutions. These two chapters prepare the way, by easy stages, 
for the formal developments in the theory of abstract groups, 
to which the remaining six chapters of Part I are devoted. 

A reader who wishes to proceed as early as possible to the 
phases of group theory presented in Part II or Part III will 
find that the prerequisites for either of the latter parts are met 
by these first two chapters of Part I, with the exception that 
also 22, 27, 48 are needed for the last half of Part II, while 
68 is needed for Part III. 

Chapter III is devoted to a development of fundamental 
theorems of abstract group theory and to the estabUshment of 
a logical connection between this theory and the theory of sub- 
stitution groups. A (1,1) correspondence between the abstract 

V 



vi PREFACE 

groups of finite order and the non-conjugate regular substitution 
groups is established in 27, where it is proved that every such 
abstract group can be represented as a regular substitution 
group and that any two simply isomorphic regular substitution 
groups must be conjugate. This (1, 1) correspondence is used 
frequently in the further development of the theory of abstract 
groups for the sake of furnishing concrete illustrations. The 
section in which it is established closes with a very simple recent 
proof of Sy low's theorem, and in 29 the interesting (^-sub- 
groups are considered for the first time in a textbook. 

The two most important categories of special abstract groups 
are doubtless the Abelian groups and the prime-power groups, 
and these two categories are treated in Chapters IV and V 
respectively. In the development of a theory of Abelian groups 
special emphasis has been placed on a determination of all their 
possible invariants, since these ' invariants seemed to offer the 
easiest means for studying various important questions in this 
theory. The Abelian groups which are groups of isomorphisms 
of cyclic groups receive especial attention in view of their appli- 
cations in number theory. Chapter V contains a considerable 
number of recent theorems. One of the most interesting of 
these is proved in 50 and establishes the fact that every poz- 
sible set of independent generators of any given prime-power 
group involves the same number of operators. 

Chapters VI and VII are more largely devoted to the 
developments due to the author than any other chapters of 
Part I. In the former of these two chapters various simple 
relations between two operators are considered and the cate- 
gories of groups which can be generated by two operators which 
satisfy these relations are determined, while the latter chapter 
is devoted to a study of groups of isomorphisms. The closing 
Chapter VIII of Part I deals with solvable groups and aims 
to be especially useful to those who seek a wide knowledge of 
the Galois theory of algebraic equations treated in Part III. 

Some of the exercises of Part I are due to questions asked 
by students and are intended to remove similar difficulties for 
the reader. In fact, this Part is based largely on a course of 



PREFACE va 

lectures given by its author at various times and the lecture 
notes were frequently changed so as to obviate difficulties which 
presented themselves to the students. Special thanks are due 
to Dr. E. A. Kircher for assisting the author in preparing these 
notes for publication, and to Professor W. A. Manning and Dr. 
Josephine E. Burns for valuable suggestions on the printer's 
proofs. 

Part II, written by H. F. Blichfeldt, seeks to give a more com- 
prehensive outline of the theory- of linear groups as developed 
up to the present moment than is contained in the pubUshed 
texts deaUng with this phase of group-theory. At the same 
time an attempt has been made to present this theory in as simple 
a manner as possible, consistent with brevity. Thus, in several 
places it has been deemed sufficient to indicate the method of 
proof of a general proposition by attending to a concrete case. 

From the outset the student is urged to work with the matrix 
form of a linear transformation ( 76). The practice thus 
gained is of great advantage throughout Part II; in particular, 
the more difficult sections of Chapter XIII will be mastered 
readily if the student has a clear mental image of the matrix 
form of the regular groups as depicted in 136 {M'). 

The introductory chapter (IX) and the chapter on binary 
groups (X) presuppose only the rudiments of ordinary group 
theory as given in 1-4, 6-9, 22, in addition to a few defi- 
nitions. By the aid of the Hermitian invariant ( 92), the 
the determination of the binary groups is here made to depend 
upon geometrical analysis, entirely with reference to Euclidean 
space. 

The following two chapters (XI, XII), exclusive of 116, 
125, are based on articles published by the author, mainly in 
the Transactions of the American Mathematical Society , 1903-1911. 
Certain proofs have been recast and new theorems added. For 
the intelligent reading of these chapters and the next following, 
the principles of 1-22, 27, 48 should be well understood. 

The somewhat difficult theory of group-characteristics (Ch. 
XIII) has been developed along fairly easy lines, differing not 
only in arrangement, but also in methods of proof, from pre- 



yiii PREFACE 

uous expositions (cf. Dickson, Annals of Mathematics, 1902; 
and the references given on p. 257). The factorization of the 
group-determinant (the determinant of M, 136; cf. Weber, 
Algebra, edition 2, vol. II, pp. 207-218) follows as an obvious 
coroUar>' to 136 (i.e., Dm^Dw) and has therefore been omitted. 

It has been found impracticable to include a discussion of 
the arithmetical nature of the elements in the matrices ( 76) 
belonging to a finite group. The student may consult Burnside, 
Proceedings of the London Mathematical Society, ser. 2, vol. 4 
(1907), p. 1 ; Schur, Mathematische Annalen, vol. 71 (1912), p. 355. 

Part III, written by L. E. Dickson, contains the essential 
principles of Galois' theory of algebraic equations (with emphasis 
on the condition for solvability by radicals), and extensive 
applications to geometrical questions. 

In the development of Galois' theory, the simpler case of 
numerical equations is treated before the case of equations 
whose coefficients involve variables, and only such rational func- 
tions are employed as are known to have denominators not equal 
to zero. 

In many of our discussions, the domain of the munbers re- 
garded as known undergoes successive enlargements; moreover, 
the initial domain is at our choice. Consequently there is an 
inherent ambiguity in the customary terms " cyclic equation," 
" simple equation," " abeUan equation," etc. An equation 
which is cyclic for one domain may not be cyclic for another 
domain. Such terms are therefore avoided in this text, being 
replaced by " equation whose group for the specified domain 
R is cyclic or simple, etc." 

For the sake of clearness, there is introduced the concept of 
solvability by radicals relatively to a domain R. The unqualified 
term " solvability by radicals " is reserved for the case in which 
the domain is that defined by the coefficients of the given 
equation. 

By the avoidance of the ambiguous terms mentioned and 
by the use of this generalization of solvability, we are able to 
, establish theorems the earlier published proofs of which were 
wholly inadequate. 



PREFACE ix 

The classic problems of the dupUcation of a cube, trisection 
of an angle, and the construction of a regular polygon of n sides 
by ruler and compasses are treated in a very simple manner by 
group theory. 

The problem of the determination of the nine points of 
inflexion of a plane cubic curs^e without singular points is treated 
adequately by group theory. The geometrical facts employed 
are not presupposed, but developed in an elementary manner. 
Similar remarks apply to the treatment of the problems of the 
determination of the 27 straight Unes on a general cubic surface 
and the 28 bitangents to a general plane quartic curve, and to 
the relation between these two problems. There is given an 
adequate basis for Hesse's and Cayley's notation for the 28 
bitangents and an elementary derivation of the perfectly sym- 
metrical notation which arose from the theory of theta func- 
tions. 

An introduction is given to a recent advance in the appli- 
cations of groups to the question of the number of real roots of 
an algebraic equation or real elements of a geometrical con- 
figuration. Without finding the actual group G of the equation 
(usually a difficult task), it often suffices to examine the sub- 
stitutions of period 2 in a group having G as a subgroup. This 
is found to be sufficient for the case of the 27 Unes on a cubic 
surface, the 28 bitangents to a quartic curve, and for an ex- 
tensive class of problems on contacts of curves, so that the 
possible numbers of real elements are found with surprising ease. 

In order that the reader may secure early a thorough acquaint- 
ance with the concept of the group of an equation, there are 
given in the first two chapters of Part III seven sets of care- 
fully selected exercises, not too difficult for the beginner, in 
addition to the numerous examples treated in the text. 

A brief, but adequate, course in Galois' theory of equations 
is provided by 1-9, 12, 13, 17, 68, 140-171, which include 
33 pages from Part I and 45 pages from Part III. 



TABLE OF CONTENTS 



PART I 

SUBSTITUTION AND ABSTRACT GROUPS 

CHAPTER I 
Examples of Groups and Fundamental Definitions 

SECTION PACB 

1. The symmetric group of order six 1 

2. The octic group 4 

3. Generating substitutions of a group 7 

4. The group of movements of plane figures 9 

5. Congruence groups 10 

6. Groups represented by matrices 13 

Exercises 15 

CHAPTER II 
Substitution Groups and Sylow's Theorem 

7. Positive and negative substitutions 16 

8. Commutative substitutions 18 

9. Transforms of a substitution and of a substitution group 20 

Exercises 23 

10. Co-sets and double co-sets 24 

11. Sylow's theorem 27 

Exercises 30 

12. Transitive groups and average number of letters in its substitutions 30 

13. Intransitive substitution groups. . . .' 32 

14. Substitutions which are commutative with each of the substitutions of a 

transitive group 35 

Exercises 38 

15. Primitive and imprimitive groups 38 

Exercises 41 

16. Groups involving no more than four letters ^ 41 

17. Simplicity of the alternating group of degree n, nj^4 43 

zi 



xii TABLE OF CONTENTS 

ncnoN , PACK 

18. Gnjups of degree five 45 

19. Holomorph of a regular group 46 

Exercises 47 

20. Class of a substitution group 47 

Eierdses 50 

CHAPTER III 
Fdndamental Definitions and Theorems of Abstract Groups 

21. Introduction 51 

22. Definition of an abstract group and a few properties of its elements 52 

23. The cyclic group 54 

Exercises 57 

24. Properties of transforms 57 

25. Construction of groups with invariant subgroups 59 

26. The dihedral and the dicyclic groups 61 

27. Representation of a group as a regular substitution group (>3 

Cayley's theorem 64 

Exercises 65 

28. Invariant subgroups and quotient groups 66 

29. Commutators, commutator subgroups and the 0-sub-groups 68 

Exercises 73 

30. Simply isomorphic groups 73 

Exercises 76 

31. Group of inner isomorphisms 76 

32. Frobenius's theorem 77 

Exercises 81 

33. Representation of an abstract group as a transitive substitution group. . . 81 
Exercises 84 

34. Historical note 84 

CHAPTER IV 
Abelian Groups 

35. Invariants 87 

36. Largest and smallest number of possible invariants 90 

37. Number of elements of a given order 93 

Exercises 94 

38. Abelian groups of given orders 94 

39. A special class of Abelian groups 95 

Exerdaes 98 

40. Sub-groups and quotient groups of any Abelian group 99 

Exercises 101 

41. Group of isomorphisms of an Abelian group 101 

42. Groups of isomorphisms of the groups of order p^ 105 

43. Abelian groups which are confomial with non-Abelian groups 107 

Exercises 109 



TABLE OF CONTENTS xiii 

SECTION PACS 

14. Characteristic sub-groups of an Abelian group 109 

Exercises 112 

45. Non-Abelian groups in which every sub-group is Abelian 112 

Exercises 114 

46. Roots of the operators of an Abelian group 114 

47. Hamilton groups 115 

Exercises 117 

CHAPTER V 
Groups Whose Orders are Powers of Prime Numbers 

48. Introduction 118 

49. Invariant Abelian sub-groups 120 

Exercises , 122 

50. Number of sub-groups in a group of order />"* 123 

Exercises 127 

51. Number of non-cyclic subgroups in a group of order p^, p>2 128 

52. Number of non-cyclic subgroups in a group of order 2"* 129 

Exercises 133 

53. Some properties of the group of isomorphisms of a group of order p^ 134 

Exercises 137 

54. Maximum order of a Sylow subgroup in the group of isomorphisms of a 

group of order p** 137 

55. Construction of all the possible groups of order p"* 138 

Exercises 141 

CHAPTER VI 
Groups Having Simple Abstract Definitions 

56. Groups generated by two operators having a common square 143 

Exercises 147 

57. Groups of the regular polyhedrons 147 

58. Group of the regular icosahedron 150 

Exercises 151 

59. Generalizations of the group of the regular tetrahedron 152 

60. Generalizations of the octahedron group 154 

Exercises ; 158 

CHAPTER Vn 
Isomorphisms 

61. Relative and intrinsic properties of the operators of a group 159 

62. Group of isomorphisms as a substitution group 160 

63. Groups of isomoq)hisms of non-Abclian groups 162 

Exercises ''^ 

04. Doubly transitive substitution groups of isomorphisms 104 



.^3gjM 



xiv TABLE OF CONTENTS 

UCnON PACB 

65. Groups of isomorphisms of the alternating and the symmetric groups.. . . 166 
Exercises 168 

66. Several useful theorems relating to groups of isomorphisms 168 

67. Group of isomorphisms of a transitive substitution group 172 

Exercises 173 

CHAPTER Vni 
Solvable Groups 

68. Introduction 174 

Exercises 177 

69. Series of composition 177 

70. Groups involving no more than one non-cyclic Sylow subgroup 181 

71. Groups whose nth group of inner isomorphisms is the identity 183 

Exercises 184 

72. Arbitrary choice of factors of composition 184 

73. Groups of order f^(^ , P and q being prime numbers 185 

74. Insolvable groups of low composite orders 186 

Exercises ., 192 



PART II 

FINITE GROUPS OF LINEAR HOMOGENEOUS TRANS- 
FORMATIONS 

CHAPTER IX 

Preliminary Theorems 

75-82. Linear transformations; product of linear transformations 193 

83-7. Groups of linear transformations 197 

88. Change of variables 203 

89. Characteristic equation 20.'> 

90. Transitive and intransitive groups 206 

91-3. Hcrmitian invariant; conjugate-imaginary groups; unitary form 207 

94. Reducible and irreducible groups 210 

95-6. Canonical form of a linear transformation and of abelian groups 212 

CHAFFER X 

The Linear Groups in Two Variables 

97-100. Introduction; geometrical analysis 215 

101-3. The groups of the regular ixilyhedra 220 

104-6. Invariants of the linear groups in two variables 224 



TABLE OF CONTENTS rv 

CHAPTER XI 

Some Spectal Types of Groups 
section paob 

106-7. Primitive and imprimitive groups 228 

108-9. Sylow groups 230 

110. On the group of similarity-transformations 233 



CHAPTER XII 

The Linear Groups in Three Variables 

111. Introduction 235 

112-3. Intransitive and imprimitive groups 236 

114-5. Groups having invariant intransitive or imprimitive subgroups 237 

116. On roots of unity 239 

117-23. Primitive simple groups 241 

124. Primitive groups having invariant primitive subgroups 251 

125. Invariants of the linear groups in three variables 253 

126. Order of a primitive group in n variables; historical note 255 

CHAPTER XIII 
Group Characteristics 

127. Introduction 257 

128-30. Number of invariants; conditions for transitivity 258 

131-2. Equivalence 262 

133-5. The sum of matrices; the product of characteristics 265 

136. Number of independent elements in the group-matrix 268 

137. Number of non-equivalent isomorphic groups 271 

138. Groups of order p'^q are composite 272 

139. Substitution groups of degree n and class n i are composite 274 



PART III 

APPLICATIONS OF FINITE GROUPS 

CHAPTER XIV 
The Group of an Algebraic Equation for a Given Domain 

140-1. Introduction, number domains 279 

Exercises 280 

142-4. Reducible and irreducible functions 280 

Exercises 282 



XVI TABLE OF CONTENTS 

SECTION PAGE 

145. Functions with ! values. . 2<S2 

14t>-7. Guloisiun resolvents . 283 

148. The group 6' of an etjuation for a domain. . . 286 

149. Characteristic proj>ertics of this proiip 6" . . 287 

150. Transitive group 289 

Exercises 291 

151. Definitions of the roots for variable coefficients. . . 292 

152. Function domains, equality . . 293 

153. Group of the general equation-. . . 294 
E.\ercises . . 29.5 

154. Rational functions belonging to a group . . 296 

155. Galois' generalization of Lagrange's theorem . . 2.)8 

156. Effect t)n the group by an adjunction to the domain. . . 299 
Exercises 300 



CHAPTER XV 

StJFFiCTENT Condition that an Algebraic Equation be Solvable by 
Radicals 

l.')7. Solvability by radicals 301 

158. Solution of a cubic equation 302 

159. Resolvent equations and their groups 303 

Exercises 305 

1(50. Equations with a regular cyclic group 306 

161-3. Cyclotomic equations, Gauss' lemma 308 

164. Sufficient condition for solvability by radicals 310 

165. Solution of a quartic equation 312 

Exercises 314 



CHAPTER XVI 

Necessary Condition that an Algebraic Equation be solvable 
by Radicals 

166. Galois' criterion for solvability -5 1 ' 

167. Theorems of Galois, Jordan, H6lder 317 

CHAPTER XVn 
Constructions with Ruler and Compasses 



168. Some celebrated problems of Greek origin. 

169. Analytic criterion for constructibility. 

170. Triscction of an angle 

171. Duplication of a cube 

172. Regular polygons 



321 
321 
323 
323 
323 



TABLE OF CONTENTS xvii 

CHAPTER XVIII 

The Inflexion Points of a Plane Cubic Curve 
section pack 

173. Homogeneous coordinates, Euler's theorem, singular points 327 

174. Hessian curve 329 

175. Inflexion points, inflexion triangles 330 

176-8. Group of the equation for the inflexion points 33:j 

179. Real points of inflexion 341 

CHAPTER XIX 

The 27 Straight Lines on a General Cubic Surface and the 28 
bitangents to a general quartic curve 

180. Existence of the 27 lines on a cubic surface 343 

181. Double-six configuration 345 

182. The 45 triangles on a cubic surface 346 

183. Group of the equation for the 27 lines 347 

184. Relation between cubic surfaces and quartic curves .'J51 

Exercises 353 

185. Steiner sets of bitangents to a quartic curve 354 

186. Notation of Hesse and Cayley for the bitangents 357 

187. Group containing the group for the 28 bitangents 362 

188. Number of real bitangents to -a. quartic curve 365 

189. Number of real lines on a cubic surface 366 

190. Actual determination of the group for the bitangents 367 

191. Symmetrical notation for the bitangents 372 

192. Further problems of contacts of curves 375 

CHAPTER XX 
MoNODROMiE Group 

193. Monodromie group M of F(z, k)=0 378 

194. M an invariant subgroup of the Galois group of F 378 

195. Applications of monodromie, differential equations 379 

196. Quintic equations, form problem 381 



PART I* 
SUBSTITUTION AND ABSTRACT GROUPS 



CHAPTER I 

EXAMPLES OF GROUPS AND FUNDAMENTAL DEFINITIONS 

1. The Symmetric Group of Order Six. There are six move- 
ments of a plane which transform into itself a given equilat- 
eral triangle situated in this plane. These are the rotations 
about the center of the triangle through angles 0, 120, 240, 
and the rotations through 180 of the plane about an altitude 
of the triangle. If we represent the vertices of this triangle 
by the letters a, b, c, the results of these six movements, which 
include the identity, are represented by the following figures: 









abb c c a b a a c 

Figs. 1-6. 

All of these figures may be obtained from any one of them by 
interchanging the letters in every possible manner. Such 
Interchanges of letters are called substitutions on these letters. 

Various symbols have been employed to represent substi- 
tutions. According to one of the oldest and most elementary 
types of symbols, the given six substitutions are represented 
as follows: 

/abc\ /abc\ /abc\ /abc\ /abc\ /abc\ 
\abc/^ \bca/' \cab/' \bac/' \acb/' \cba/' 

This part was written by (J. A. Miller. 



2 EXAMPLES OF GUOUPS; DEFINITIONS (Ch. I 

This notation implies that each letter is to be replaced by the 
one just below it in the same symbol. The same substitutions, 
in order, are commonly represented by the following briefer 
symbols: * 

1, ahc, ach, ah, he, ac. 

This notation implies that each letter is to be replaced by the 
one which follows it in the same symbol, the last being replaced 
by the first. Letters which are not replaced are omitted in 
this notation, and the symbol for unity is used to represent 
the identity; that is, the substitution in which every letter is 
replaced by itself. 

It is easy to verify the fact that any two of these substitu- 
tions, when performed successively, are equivalent to a single 
one of them. For instance, if we first apply ah and then ac 
the result is the same as if we had applied ahc only once. The 
process of combining (composing) two substitutions into one is 
called multiplication, and it is denoted by the common symbols 
for multiplication. Hence ahc is said to be the product of ab 
and ac. Since ac-ah = acb, and ab-ac = abc, it results that the 
commutative law of multiplication is not always satisfied as 
regards the multiphcation of substitutions. 

A set of distinct substitutions, which has the property that 
no additional substitution can be obtained by multiplying 
successively each substitution of the set into all the substitu- 
tions of the set, is called a substitution groups Hence the given 
set of six substitutions constitutes a substitution group. The 
number of the distinct substitutions of a group is called the 
order of the group and the number of the distinct letters in its 
substitutions is the degree of the group. The totality of the 
possible ! substitutions on n letters evidently constitutes a 

* These symljols have been called the normal forms of substitutions, J. de 
S^uicr, Croupes de Substitutions. 1912, p. 3. They are often inclosed in 
parentheses. 

t The term group in this technical sense is due to E. Galois (1811-32). The 
statement that the term group was not used before 1870 with its present technical 
meaning, which is found in the Encydopadia Britannica, eleventh edition, 
vol. 22, p. 620, is incorrect. 



S ll SYMMETRIC GROUP OF ORDER SIX 3 

substitution group, and this is known as the symmetric group 
of degree . 

The symmetric group of degree n exists for every value 
of , > 1, and it includes every possible group on these letters. 
There may, however, be other groups on n letters. In fact- 
it is easy to verify that the three substitutions 

1, abc, acb 

constitute a second group on the three given letters. Hence 
we have found two substitution groups on three letters, and 
it can be verified that no other groups involving these three 
letters are possible. That is, if a set of substitutions involving 
the three letters a, b, c constitutes a group, this group is of 
order 3 or of order 6, and there is only one such group of each 
of these orders if we regard two substitution groups identical 
when they differ only as regards the letters involved. This 
group of order 3 could have been found by trial combinations 
of the six possible substitutions on three letters, but it results 
also directly from the fact that it corresponds to the rota- 
tions of the given triangle about its center through the angles 
0, 120 and 240". 

As this group of order 3 is contained in the symmetric group 
of order 6 it is called a subgroup or a divisor of the latter. The 
identity is a subgroup of every group. The only other sub- 
groups of this symmetric group are of order 2. There are three 
such subgroups, viz., 

1, ab; 1, ac; 1, be. 

Hence the symmetric group of order 6 has four subgroups besides 
the identity. The last three of these correspond to the groups 
of movements through the angle v around the lines of sym- 
metry of the given triangle. 

The symmetric group of order 6 could also have been found 
by considering the possible permutations of three variables 
which do not alter a symmetric function of these variables. 
A simple instance of such a function is the following: 

x-\-y+z. 



4 EXAMPLES OF GROUPS; DEFINITIONS (Ch. I 

The given group of onler '6 may be obtained from a considera- 
tion of the permutations which leave formally unaltered the 
function 

(x-y){y-z)(z-x) 

of the independent variables x, y, z. 

2. The Octic Group.* There are eight movements of a 
plane which transform into itself a square situated in this 
plane. If we represent the vertices of this square by the letters 
a, b, c, d, the results of these eight movements are represented 
by the following figures: 

d c ha ad c h 

a b c d be do, 



d 


a 


b 


c c 


d 


a 


b 


c 


b 


a 


d b 
Figs. 7-14. 


a 


d 


c 



The corresponding eight substitutions are as follows: 

1, ac-bd, abed, adeb, ae, bd, ab-cd, ad- be. 

It should be observed that three of these substitutions are 
separately the products of two substitutions on distinct sets 
of letters. For instance, the substitution ae-bd "mplies that 
a and c are replaced by each other, and that b and d are also 
replaced by each other. 

The symmetric group on four letters is composed of 24 
substitutions. Hence the octic group is composed of one- 
third of the possible substitutions on four letters. It con- 
tains three subgroups of order 4, viz., 

1, ac-bd, abed, adeb; 1, ac-bd, ac, bd] 

1, ac-bd, ab-ed, ad-be. 

The first of these corresponds to the rotations about the center 
of the square through the angles 0, 7r/2, x, and 37r/2. The 

This group is also known as the group of the square, and as the dihedral 
group of order 8. It is given in 108 of the noted article by J. L. Lagrange 
entitled, " Reflexions sur la resolution algebrique des Equations,' which was 
published in the Nouveaux Mimoires de VAcadlmk royale de Berlin, 1771. 



21 OCTIC GROUP 6 

second corresponds to the movements through the angle v 
around the diagonals, and their combinations. The third cor- 
responds to the movements through the angle x around the 
lines joining the middle points of opposite sides, and their 
combinations. 

The three subgroups of order 4 have the subgroup 1, acbd 
in common. There are four other subgroups of order 2 in the 
octic group, viz., 

1, ac; 1, bd; 1, ab-cd; 1, adbc. 

Hence the octic group has three subgroups of order 4 and five 
subgroups of order 2 besides the identity. It can be verified 
that no other subgroup exists in this octic group. 

Another illustration of this group may be based on the 
function of four variables 

which is transformed into itself by the following eight sub- 
stitutions 

1, X\Xz'X2PC4^, XiX2X:iX4^, ^^1X4^^3X2, X\Xz, X2X4, X\X2'X3Xa, X\Xa' X2X:i. 

These substitutions constitute the octic group, since they can 
be changed or transformed into those of the given octic group 
by the substitution 

x\a-X2b-xzC-XAd. 

Substitutions and substitution groups which can be trans- 
formed into each other by an interchange of letters are said 
to be conjugate. 

The remaim'ng sixteen substitutions on these four variables 
transform the function xiXi-{-X2X\ into one of the following 
two functions : 

X\X2-\rXzXA, XiXJk-\-X2XZ' 

Each of these two functions belongs to the group of all the 
substitutions on these four letters which transform the func- 
tion into itself. These two groups are distinct from each 
other and also from the octic group obtained above. In fact, 



6 



EXAMPLES OF GROUPS; DEFINITIONS 



(C'h. I 



the three groups are' conjugate,* and they have in common 
the following subgroup of order four: 

1, X\X2'XzXa, X\Xz'X2XAy X\Xa' X2Xi. 

Hence the symmetric group of degree four contains three con- 
jugate octic groups. These octic groups have four substitutions 
in common, and therefore they involve sixteen distinct substi- 
tutions. 

An interesting illustration of the applications of the octic 
group in elementary trigonometry is furnished by the opera- 
tions of finding the complement and the supplement of an angle 
a, and of the angles obtained from a by these operations. We 
thus obtain the following eight geometric angles and no more: 

a, 90 -a, 180 -a, 90 +a, 270 -fa, -a, 270 -a, 180 +a. 




270-a 



270+a 



Fig. \T>. 

* Two ronjugalc substitution groups are often regarded as the same group, 
but when they arc subgroups of a given group they arc often said to be distinct 
whenever one contains at least one substitution which is not in the other. In 
fisting the possible substitution groups on a given number of letters conjugate 
groups arc regarded as identical. 



3] GENERATING SUBSTITUTIONS 7 

Each of these eight angles may be regarded as representing also 
the operation by means of which it can be derived from a. From 
this point of view these angles constitute a group of order 8. 
That this is actually the octic group may be seen as follows: 

If the angle a is represented by a point on a unit circle, 
the process of finding the complement of a is equivalent to 
reflecting a on the bisector of the first and third quadrants, 
since the geometric meaning of subtraction is a reflection on 
the point midway between and the minuend. Hence the 
process of finding the supplement of any angle a is equivalent 
to reflecting on the Y-axis the point representing a. Since these 
two kinds of reflections transform into itself the square whose 
sides are parallel to the coordinate axes, inscribed in the unit 
circle, it results that the group represented by the eight angles 
which may be obtained from a by the operations of finding 
the complement and the supplement is the octic group. 

One important difi^erence between the group of movements 
of the triangle and the group of movements of the square should 
be emphasized here. In the former case we obtain exactly the 
same substitution group on the letters a, b, c no matter how 
these letters are arranged so as to represent the vertices. In 
the latter case, some possible rearrangements give rise to a differ- 
ent substitution group. In fact, by starting with different 
arrangements we may obtain three possible conjugate octic 
groups on the letters a, h, c, d by movements which transform 
the figure as a whole into itself. We thus obtain another ele- 
mentary illustration of the important concept of conjugate 
substitution groups, or of subgroups conjugate under the 
symmetric group. 

3. Generating Substitutions of a Group. The different 
distinct powers of any substitution constitute a group, which 
is called cyclic. If it is not possible to find at least one substi- 
tution in a group such that all the others are powers of it, the 
group is non-cyclic. The s^-mmetric group of order 6 and the 
given conjugate octic groups are non-cyclic. Each of the latter 
groups involves one cyclic and two non-cyclic subgroups of 
order 4. 



8 EXAMPLES OF GROUPS; DEFINITIONS [Ch. I 

A substitution whose powers give all the substitutions of a 
group is said to generate this cyclic group, and the order of 
this cyclic group is equal to the order of the substitution. It is 
always possible to select such a generating substitution in 
more than one way except when the order of the cyclic group 
is 1 or 2. For instance, the cyclic group of order 4 

1, ac-hd, abed, adcb 

is generated by abed as well as by adcb, and the cyclic group 
of order 3 

1, abc, aeb 

has also two generating substitutions. 

In general, a substitution group is said to be generated by 
a set of substitutions provided all of the substitutions of the 
group can be obtained by combining those of the set. The least 
number of substitutions that can generate a non-cychc group 
is two. Each of the non-cyclic groups which have been con- 
sidered thus far can be generated by two of its substitutions. 
For instance, the symmetric group of order 6 can be generated 
by any one of its three possible pairs of two distinct substi- 
tutions of order 2. It can also be generated by any one of the 
six distinct pairs composed of one substitution of order 2 and 
one of order 3. Hence the symmetric group of order 6 has 
nine distinct pairs of generating substitutions. 

The octic group cannot be generated by every possible pair 
of distinct substitutions of order 2, since some such pairs gen- 
erate only four substitutions. In fact, it is easy to verify that 
only four out of these ten possible pairs generate this group, 
while each of the remaining six generate a group of order 4. 
The square of the substitutions of order 4 cannot be used as 
one of a pair of generating substitutions of the octic group, 
but every other substitution besides the identity of this group 
occurs in such a pair. Hence it is not difficult to verify that 
there are exactly 12 possible pairs of generating substitu- 
tions of the octic group. 

Any set of substitutions on n letters generates some sub- 
stitution group on these letters, which is contained in the sym- 



4) GROUPS OF PLANE FIGURES 9 

metric group of degree n. It is not necessary that each of 
these substitutions should actually involve all of these n letters. 
For instance, it may be found by trial that the two sub- 
stitutions abc and abd generate a group of order 12, while the 
two substitutions abc and ad generate the symmetric group of 
order 24. There is no upper limit for the order of a group which 
can be generated by two substitutions if these substitutions 
be chosen arbitrarily and their degrees are not limited. 

A set of X substitutions Si, S2, . . . , Sx oi a. finite substi- 
tution group G is called a set of generators of G provided there is 
no subgroup in G which includes each of these substitutions. 
When these substitutions satisfy the additional condition that 
G can be generated by no X 1 of them, the set is said to be a 
set of independent generators of G. Such a set can usually 
be chosen in many different ways. 

4. The Groups of Movements of Plane Figures. The sym- 
metric group of order 6 and the octic group are special cases of 
the groups of movements of regular polygons. The regular 
polygons of n sides are evidently transformed into themselves 
by the cyclic group of order n which is generated by the sub- 
stitution corresponding to the permutation of the vertices 
when the polygon is rotated around the center through the 
angle 2ir/. They are also transformed into themselves by n 
substitutions of order 2 which correspond to the permutation 
of the vertices when the polygons are rotated successively 
through the angle x around their different lines of sjTnmetry. 
As no other movements transform these polygons into them- 
selves, it results that the group of movements of a regular 
polygon of n sides is of order 2. 

According to a common definition of regular polygons there 
is only one regular polygon of 3, 4, or 6 sides, but there 
are two regular polygons of five sides, as may be seen by 
connecting alternate vertices, and there are three such poly- 
gons of 7 sides. In fact, it is not difficult to see that the num- 
ber of such regular polygons of n sides is equal to one-half the 
number of generating substitutions of the cyclic group of order 
. All of these regular polygons of n sides belong to the same 



10 EXAMPLES OF GROUPS; DEFINITIONS (Ch. 1 

substitution group, but this group can be most easily studied 
by means of the polygon obtained by connecting successive 
points dividing the unit circle into n equal parts. It contains 
n or + l substitutions of order 2 according as n is odd or even. 

If we consider the general problem of transforming a system 
of n, n>2, non-collinear points in the rigid plane among them- 
selves, it is important to observe that all such transformations 
leave a given point invariant, viz., the center of the smallest 
circle that circumscribes the system of points in question. 
Hence the possible movements are restricted to rotations around 
this center, or rotations through the angle tt on a line passing 
through this invariant point. The former rotations must 
constitute either exactly half or all of the possible movements. 
In the case of the general parallelogram they evidently con- 
stitute all of the possible movements, while they constitute 
exactly half of the possible four movements in the case of the 
general rectangle. This result may also be expressed as follows: 
A plane figure either has no line of symmetry or it has as many 
lines of symmetry as it admits different plane rotations around 
the center of its smallest circumscribing circle. In the former 
case its group of movements is cyclic, while it is non-cyclic 
in the latter case. 

5. Congruence Groups. It will be proved later that every 
finite group can be represented as a substitution group. Many 
groups present themselves naturally in different forms and 
hence it is desirable to study groups represented in different 
ways. For the present we shall, however, study these groups 
by means of substitution groups. 

Suppose that the first m l p)Ositive integers, together with 
zero, are combined by addition, and the sums are replaced by 
their least positive or zero residues modulo tn. It is clear 
that no new numbers are obtained in this way. If any number 
is added separately to itself and to each of the others, the tn 
numbers will be permuted according to a substitution which 
may be associated with this added number. The substitutions 
which are thus associated with all these numbers constitute 
the cyclic group of order m. 



5] CONGRUENCE GROUPS . 11 

The substitution which is associated with is the identity, 
while the one which is associated with 1 must be a generating 
substitution of the cyclic group of order m. If k is any one of 
these m numbers, the substitution which corresponds to k 
will be of order m/d, where d is the highest common factor 
of m and k. Two substitutions whose product is the identity 
are said to be the inverses of each other. Hence the inverse 
of the substitution which corresponds to k is the one which 
corresponds to m k. 

Moreover, if the (f>{m)* positive integers which are prime 
to m and not greater than m are combined by multiplication, 
and the products are reduced mod ulo m, no additional numbers 
are obtained by this operation. For instance, if w = 8 and 
we multiply the four numbers -^ 

1, d, 5, 7 

in succession by 1, 3, 5, 7, we obtain the following non-cyclic 
substitution group of order 4: ^rr. urv v;>()W 

1, 13-57, 15-37, 17-35 

If OT = 5 and we combine the numbers 1, 2, 3, 4, by multiplica- 
tion, there results the following cych'c group of order 4: 

1, 1243, 1342, 14-23 

The two given types of groups furnish illustrations of the 
following very important category of congruence groups, where 
p is a prime number, viz., the groups formed by all the f)ossible 
linear substitutions of the following form: 

X =ax-\-b (mod PH , ^ . ^ ^ , 

6 = 0, 1, 2, ... , p-l. 

* This symbol was first used by Gauss to represent the present concept. It 
is called the lolietU of m according to Sylvester. The French called it the 
indicator of m, and the Germans commonly call it the FmIct ^-function of m. 

t This group is sometimes represented by the following general substitution: 



/O I 2 ... ^ - 1 \ 

\b a+b 2a+b . . . (Jt-l)a+b/ 



12 EXAMPLES OF GROUPS; DEFINITIONS [Ch. I 

It is easy to find the oMer of such a substitution as follows: 

Let 5 represent the substitution x'=ax-\-b. 
Then s^ is x' = a^x-\-ia-\-l)b, 



and s'isx'=a'x+(a'-i+a'-2+ . . . +a + l)6. 

If s' = l, it is necessary that a'"=l (mod p). This is equiv- 
alent to 

a'-l = (a-l)(a'-^+a'-^+ . . . +a + l)=0 (mod />). 

When a^l (mod p), it results that a necessary and suf- 
ficient condition that 5'" = 1 is that a^ = l (mod p). That is, 
the order of s is the exponent to which a belongs (mod p) 
except when a = l (mod p). In the latter special case, j is 
clearly of order p when b is not zero. 

It should be observed that the order of 5 is independent of 
the value of b except when a = \ (mod p). Hence the con- 
gruence group under consideration has p1 substitutions 
of order p, and p6{d) substitutions of order d, d being any divisor 
of p l, including /> 1 but excluding 1. The order of this 
group is therefore p{p l), as is also evident from its definition. 
It is commonly known as the mclacyclic group of order p{p \). 
The term metacyclic has also been used by H. Weber in his 
Lehrbuch der Algebra, 1895, page 598, to define a more general 
category of groups, but the definition given here is the older 
one and is still commonly used. 

A particular type of linear groups considered by E. Galois 
consists of the w* operations which may be represented by 

I 2i, 22, . . . ,zt zi-f-ci, 22+C2, ., Zt-\-Ct \ (mod m) 

where each 2< is replaced by Zt-{-Ci (mod m). In a more 
explicit notation, we have 

z'i=Zt-\-Ci (mod m), i=l, 2, . . . , k. 

This group of order w* was called by A. L. Cauchy the group 
of arUhmetic substiiulions. In particular, when m = k = 2 we 



S6] 



GROUPS REPRESENTED BY MATRICES 



18 



obtain the non-cyclic group of order 4 noted above. This 
general group is clearly generated by the k substitutions of 
which the ith (for i = 2, 3, . . . , ife-1) is 

z'i=zi, . . .,z\-i=Zi-i,z't = Zi-\-l,z\+i=Zt+i, . . .,2't=Z4(modm). 

For i=l and i = k this substitution becomes respectively 

z'l=Sl + l, 2^2=22, . . ., 2't = Zt 

z'i = 2i, . . ., z'_i = 2t_i,2't = 2t4-l (mod m). 



and 



Each of these k substitutions generates a cyclic group 
of order m, and the entire group is said to be the direct product 
of these k independent cycHc groups. 

6. Groups Represented by Matrices. If two matrices of order 
n are multiplied together in the ordinary manner * there results 
a matrix of order n. It may happen that the elements of the 
matrices are of such a nature that only a fim'te number of 
different matrices result when a given set of them are com- 
bined by multiplication in every possible order. For instance, 
the six matrices 



1 




-1 1 




-1 




1 




-1 




1 -1 


1 


} 


-1 


> 


1 -1 


> 


1 


J 


-1 1 


> 


-1 



and the six substitutions 

1, abc, acb, ah, ac, be 

may be put into a (1, 1) correspondence in the given order so 
that hke products always correspond. That is, the given 
matrices combined under multiplication constitute the sym- 
metric group of order six, or the group of an equilateral triangle. 
We thus meet this group in another very important and exten- 
sive field of mathematics, viz., the theory of matrices. It 
may be observed that the. first three of these matrices con- 
stitute the subgroup of order three and they can therefore be 
put into a (1, 1) correspondence, as regards multiplication, 
with the cube roots of unity. 

* For a definition of the product of two matrices cf. BAcher, Inirodudum t9 
Higher Algebra, 1907, p. 62. 



14 



EXAMPLES OF GROUPS; DEFINITIONS 



[Oh. 1 



It is not difficult to verify that the following six matrices 
also constitute a group of order six as regards multiplication: 



1 
1 


> 


-1 1 

-1 


> 


-1 

1 -1 


> 


-1 0| 
0-1 ' 


1 -1 

1 


> 


1 

-1 1 



This group of order six can, however, not be put in a (1, 1) 
corresp>ondence with the one considered in the preceding para- 
graph, although the first three matrices are the same in the 
two groups. The present group of order six is generated by a 
single matrix and hence it is cyclic. As a generating matrix 
we may use either one of the last two matrices. It may also 
be observed that multiplication is commutative in the present 
group, while this is not the case in the group of the preceding 
paragraph. The matrices of the present group can be placed 
in a (1, 1) correspondence with the six sixth roots of unity as 
regards the operation multiplication. 
The four matrices 



1 
1 


) 


-1 
0-1 


> 


1 
0-1 


> 


-1 
1 



evidently serve as another illustration of the non-cyclic group 
of order 4 when they are combined by multiplication, while 
the four matrices 



1 
1 


> 


-1 
0-1 


> 


1 
-1 


> 


-1 

1 



combined according to the same operation, constitute the cyclic 
group of order 4. As an instance of matrices which constitute 
a cyclic group of an arbitrary order we may observe that the 
matrices 



1 
1 



1 1 
1 



1 2 
1 



1 3 
1 



1 m-1 
1 



constitute the cyclic group of order m when they are combined 
as to multiplication, and the elements are reduced modulo m. 



61 GROUPS REPRESENTED BY MATRICES 16 



EXERCISES 

1. In a letter to C. Hennite [Paris Comptes Rendus, vol. 49 (1859), 
p. 115], E. Betti states that the following substitutions: 

z'=4z, 2'=-, 2'=3^ (mods) 
t s 1 

generate a group of order 12. Verify this statement, and prove that the 
first two of these substitutions generate the non-cyclic group of order 4. 
Either of the first two of the three given generating substitutions could be 
omitted without affecting the resulting group. 

2. Prove that the transformations of the form z' = (mod 3), o, 

yz+i 

/3, 7, 5 being integers such that aS ^=1 (mod 3), constitute a group of 
order 12 which has the same number of elements of each order as the one 
in Example 1. Cf. F. Klein, Mathematische Annalen, vol. 14 (1879), p. 418. 

3. By subtracting a given number n from unity and dividing unity 
by n, and then performing the same operations successively on the result- 
ing numbers, it is possible to obtain, in general, six different numbers. 
The only exceptions occur when n has one of the following 8 values: 1, 
2, ; 1,0, oo; ^(1V 3). If n represents an anharmonic ratio of four 
points, then the six values of n obtained by the given operations represent 
all the values of this ratio. 



CHAPTER n 
SUBSTITUTION GROUPS AND SYLOW'S THEOREM * 

7. Positive and Negative Substitutions. The symbol aia2 
... On is called a cyclic suhslitulion of degree n, and it implies 
that each of these letters is replaced by the one which follows 
it, and that is replaced by ai. That is, it implies a cyclic 
interchange of its letters, and any one of these letters may be 
written first without changing the rrieaning of the substitu- 
tion. Any permutation of a set of m distinct letters can evi- 
dently be effected by one or more cyclic substitutions on these 
letters. If a substitution consists of more than one cycle it 
is said to be non-cyclic and its various cycles are usually sepa- 
rated by periods, and the number of letters in all these cycles 
is the degree of the substitution. For instance, a\a2az-aAas 
denotes a substitution of degree 5 which implies a cyclic inter- 
change of the letters ai, 02, as in order, and also an interchange 
of the letters a^, as- 

A cyclic substitution which involves only two letters is 
caUed a transposition. The fact that every possible substi- 
tution is a product of transpositions results directly from each 
of the following equations: 

aia2a3 . . an = aia2'aia3' . . . -aian 

aiChi-Onan-i- . . . a3a2 

A given substitution can be factored into transpositions in an 
infinite nunr^r of different ways, but the number of the 
transpositions intQ which a particular substitution can be 

C i 
The general theory of substitutions and substitution groups is developed 
from the beginning in the present chapter. A few definitions given in the pre- 
ceding chapter are, however, not repeated. 

16 



7] POSITIVE AND NEGATIVE SUBSTITUTIONS 17 

factored is either always even or it is always odd. We pro- 
ceed to prove this important elementary theorem. 

Let 5 represent any substitution on m letters and suppose 
that ^ has been factored into transpositions in various ways. 
Each of these transpositions changes the sign of the following 
determinant: * 



A = 



1 ai ai^ . . . ai*^~^ 



which is not identically zero. As the various sets of tran- 
positions which are equivalent to s must have the same effect 
on A as 5 has, it results that the number of transpositions in 
every set is odd if 5 transforms A into A, and this number is 
even if 5 transforms A into itself. 

A substitution is said to be positive if it can be factored 
into an even number of transpositions. If it can be factored 
into an odd number of transpositions it is called negative. For 
instance, abc = ab-ac is positive, and abed = ab ac ad is nega- 
tive. The product of two positive substitutions is positive 
and the product of two negative substitutions is also positive. 
Hence a product of a set of substitutions is positive or negative 
according as it involves an even or an odd number of negative 
substitutions. 

If a substitution group involves a negative substitution 
then exactly one-half ^of its substitutions are negative. In 
fact, if all its positive substitutions are multiplied into this 
negative substitution, all of these products are distinct and 
negative. Hence it has at least as many negative substitutions 
as positive ones. On the other hand, if this negative substi- 
tution is multiplied into all of its negative substitutions, all 
these products are distinct and positive. Hence it has at 
least as many positive substitutions as negative ones. In other 

* This determinant is known as the determinant of Vandermonde or o( 
Cauchy. It is equal to the product 

Uioi-qt); i, k=l, 2, . . . , m; i>k. 



18 SUBSTITUTION GROUPS [Ch. II 

words, either all the substitutions oj a substitution group are posi- 
tive or exactly half oj tliem are positive. 

W^henever a group contains negative substitutions it con- 
tains a subgroup of half its own order, composed of its positive 
substitutions. In particular, the symmetric group of degree 
n contains a subgroup of order !/2 which is composed of its 
positive substitutions. This subgroup is called the alternating 
group of degree n. Hence there are at least three distinct 
groups of degree n whenever w>3, viz., the group generated 
by a cyclic substitution on n letters, the alternating group, and 
the symmetric group. It will be proved that other groups of 
degree n exist for every value of w>3. The cyclic substi- 
tution of degree^ is positive or negative according as n is odd 
or even. 

The product of two transpositions which have a common 
letter is always of the form abc. A positive cyclic substitu- 
tion is always the product of substitutions of the form abc, 
since it is the product of an even number of transpositions 
having a common letter. A substitution composed of two 
negative cyclic substitutions is also the product of substitu- 
tions of the form abc. In fact, such a substitution may be 
regarded as the product of two distinct sets of transpositions 
such that all the transpositions of each set have a common 
letter and such that each set involves an odd number of trans- 
p)ositions. Hence it remains only to observe that a substitu- 
tion composed of two transpositions having no letter in common 
is the product of substitutions of the form abc. This fact 
results directly from the pro'duct 

ab-cd = acb bdc. 

That is, every possible positive substitution is the product of ^sub- 
stitutions of the form abc. 

8. Commutative Substitutions. Let s and / represent two 
substitutions. If st = ts, these two substitutions are said to be 
commutative. For instance, il s = ab-cd and t = acbd, it is easy 
to verify that st = ts = adbc. On the other hand, if s = ab-cd 
and t\=bc, it results that st\=acdb while tis = abdc. Hence 



8] COMMUTATIVE SUBSTITUTIONS 19 

the two substitutions ab-cd, acbd are commutative, but ab-cd 
is not commutative with be. Two substitutions which have 
no letter in common are always commutative. 

It is often necessary to find all the substitutions on certain 
letters which are commutative- with a given substitution. The 
solution of this problem is based on finding all the substitutions 
on the letters ai, a2, . . . , an which are commutative with 
the cyclic substitution Si=aia2 ... On- It is clear that Si 
is commutative with all of its powers, and hence Si is commu- 
tative with at least n substitutions, including the identity, 
on the letters ai, a2, . . . , On- 

All substitutions which are commutative with si must also 
be commutative with Si"\ where ai is any positive integer. 
Suppose that /2 is a substitution on the letters ai, a2, . . . , 0% 
which is commutative with si but is not a power of Si. It is 
evident that /2 must involve each of the letters ai, a2, . . . , a. 
Hence we may suppose that h = CLiaa , where a is one 
of the numbers 2, 3, . . . , w. Since 5i"~^=aiaa . . . , it 
results that t2Si*'^^~'' is a substitution which is commutative 
with 5i, does not involve ai, and is not the identity. That is, 
we arrive at an absurdity by assuming that more than n sub- 
stitutions on the letters ai, a2, . . . , On are commutative with 
si. This proves the theorem: The only substitutions on n letters 
which are commutative with a cyclic substitution on these letters 
are the powers of this cyclic substitution. 

If a substitution 52 is composed of X cycles such that no 
two of these cycles involve the same number of letters, then 
all the substitutions on the letters of S2, which are commutative 
with S2 must also be commutative with each cycle of s-y. The 
number of the substitutions which are commutative with 50, 
and involve only letters contained in 52, is therefore equal to 
the product of the orders of the cycles of 52. For instance, 
the substitutions which are commutative with the following 

substitution 

abcde-fgh, 

and involve only its eight letters, constitute a substitution 
group of order 15. 



20 SUBSTITUTION GROUPS [Ch. II 

A substitution 53 which involves k equal cycles is commuta- 
tive with substitutions which permute these cycles according 
to the symmetric group of degree k. If each of these cycle- 
involves n letters, the substitutions on these kn letters which 
are commutative with S3 must therefore constitute a substi- 
tution group of order n^-kl. For instance, the substitutions 
on the nine letters involved in the following substitution 

abc'def-ghi, 

and which are commutative with this substitution, constitute 
a group of order 27-6 = 162. 

9. Transforms of a Substitution and of a Substitution Group. 
If s and / represent any two substitutions, it is possible to 
find a third substitution by means of the operation s~Hs = ti, 
where s~^ represents the inverse of s. The substitution /i 
is called the transform * of / as regards s. There is a very 
simple rule for deriving /i if s and / are given. We proceed to 
develop this rule. 

Suppose that 

t= . . . aap ... 

It is not assumed that a^ or a^ actually appears in s when s is 
written in the normal form, since a'^ may be identically equal 
to da and a'fi may be identically equal to a^. Hence the given 
notation is entirely general. It is easy to see that 

t\= . . . dad p . . 

That is, to obtain the transform of t as regards s we simply replace 
each letter in t by the otie by which s replaces this letter. f For 
instance, if seabed- ef and t = ab-ce, then s~Hs = bc-df. In 

This transformation is fundamental in the theory of groups. Many of 
its properties were developed by E. Betti in 1852; Annali di Scienze malematichc 
efisiche, vol. 3, p. .'.'). 

t This simple method to find the transform of a substitution is found in C. 
Jordan's thesis, Paris, 1860, p. 14. 



9] TRANSFORMS OF SUBSTITUTIONS 21 

particular, if a substitution contains a certain number of cycles 
of a given order, each of its transforms contains the same 
number of cycles of this order. 

A necessary and sufficient condition that t = ti is that s 
and t are commutative, since the equation s~Hs = t implies that 
ts = st and conversely. Each of two commutative substitutions 
is transformed into itself by the other. If a substitution is 
transformed into itself by all the substitutions of a group it 
is said to be invariant under this group. When this substitu- 
tion belongs to the group it is said to be an invariant substitu- 
tion of the group. The identity is invariant under every group 
and is an invariant substitution of every group. As an instance 
of another invariant substitution it may be observed that 
ac-bd is invariant under the following octic group : 

1, ab-cd, aC'bd, ad- be, ac, bd, abed, adeb. 

The symmetric group of order 6 contains no invariant substi- 
tution besides the identity. 

If each substitution of a group G is transformed by means of 
a substitution s there results a group G\ which is called the 
transform of G with respect to s, or the conjugate of G with 
respect to s, and is represented by the symbol 

s~^Gs = Gi. 

Conjugate substitutions and conjugate groups are also called 
similar. When Gi=G the substitution 5 is said to transform G 
into itself, and G is said to be invariant under s. Every group 
is invariant under its own substitutions. A subgroup which 
is invariant under all the substitutions of a group is called an 
invariant, or self -conjugate, subgroup. If a group involves 
negative substitutions all of its positive substitutions consti- 
tute an invariant subgroup. In particular, the alternating 
group of degree n is an invariant subgroup of the symmetric 
group of this degree. 

Suppose that s is transformed into itself by some but not 
by all of the substitutions of a group G. The substitutions 



22 SUBSTITUTION GROUPS [Ch. II 

of G which transform s into itself form a subgroup // of G. 
If / is any substitution of G which is not in H then all the sub- 
stitutions obtained by multiplying / on the left by a substi- 
tution of // will transform s into the same substitution. For, 
if /i is any substitution of H, we have the equation 

Moreover, if t'-^st' = r^st, it results that s = t'r^sU'~K That 
is, U'~^ is in H. Hence its inverse t't~^ is also in H. That is, 
/' is one of the substitutions obtained by multiplying t on the 
left by some substitution of H. 

From what precedes it results that all the substitutions of 
G can be divided into equal sets such that each set is composed 
of all the substitutions of G which transform s into the same 
substitution. The totality of the substitutions into which 
s is transformed under G forms a complete set of conjugates of 
s under G. The number of different substitutions in a com- 
plete set of conjugates under a group is a divisor of the order 
of the group. A complete set of conjugate subgroups under G 
is defined in a similar manner. 

Incidentally we proved above that the order of G is divisible 
by the order of its subgroup H. We proceed to prove that the 
order g of G is divisible by the order k of any subgroup K* 
In fact, all the substitutions of G can be written in the form of 
a rectangle, whose first line is composed of the substitutions 
of K, as follows: 

1, 52, ^3, . ' ., St 
/2, ^2^2, Szt2, . . ., Stt2 



t\, S2t\, 53/x, . . ., Stt\ 



In this rectangular array the substitutions t2, . . . , i\ are any 
substitutions of G which do not occur in any of the preceding 
rows. 

* If a group IS represented by a capital letter the corresponding small letter 
usually represents the order of the group. 



9] TRANSFORMS OF SUBSTITUTIONS 23 

To prove that ^ is a divisor of g it is only necessary to prove 
that no substitution of G can occur twice in such an array. 
This fact can be easily proved as follows : 

If sjy = Sfity {a, ^%k), th en 5 = 5^ ; 
and 

if sJy = Spti{b>'y)j then Sfi~^sJy=Sa'ty = ti{a%k). 

Hence it results that no substitution of G can occur twice in 
such an array and we have established a fundamental theorem, 
known as the theorem of Lagrange, which may be stated as 
follows : 

The order of a group is divisible by the order of each one of its 
subgroups. 

This may be regarded as the most important theorem of 
group theory. It will appear that a subgroup has properties 
similar to those of a modulus in number theory. Hence a sub- 
group is sometimes called a modulus of the group. The 
quotient obtained by dividing the order of a group by the 
order of a subgroup is called the index of this subgroup under 
the group. The index of a subgroup is therefore always a 
positive integer. 

EXERCISES 

1. The order of a group is divisible by the order of each of its sub- 
stitutions. (Cauchy.) 

2. The order of every substitution group on n letters is a divisor of !. 

(Cauchy.) 

3. All the substitutions which are common to two groups constitute 
a group. This is known as the cross-cut of these two groups. 

4. The number of substitutions in a complete set of conjugate sub- 
'stitutions under a group cannot exceed the quotient obtained by dividing 

the order of the group by the order of one of these substitutions. 

5. The symmetric group of degree n, n>2, does not contain any invari- 
ant substitution besides the identity. 

6. The alternating group of degree n, >3, does not contain an invariant 
substitution besides the identity. 

7. Find the following products: 

abcdeXbdc= , ac-bdxbcfe= , 

"^ adeXabcXcicbed= , acXcdXdc 



24 SUBSTITUTION GROUPS ICh. II 

8. Find a value of / in the form of a product of transpx)sitions such that 
abcdefg =aeXafXagXbftXekXcdXt. 

9. Transform ace-gf by hef, and also by ab-cd. 

10. Write the 12 positive substitutions on four letters, and find the 
groups composed of all the substitutions on these four letters which 
transform into itself each of these 12 positive substitutions. 

10. Co-sets and Double Co-sets. From the arrangement of 
all the substitutions of a group in the form of a rectangle in 
which the first line is composed of all the substitutions of a 
subgroup (9), it results directly that, if H is any subgroup of 
the group G, all the substitutions of G can be written in the 
following form : * 

G=H-\-nt2-{- . . . -\-ntx. 

In this notation, Hta{a = 2, . . . , X) stands for all the products 
formed by multiplying every substitution of H into /. The 
sets of substitutions represented by Hta are called co-sets f of 
G as regards H. It is important to observe that / may be 
replaced by any one of the substitutions of the co-set to which 
it belongs without changing the co-set. In other words, if 
two such co-sets have one substitution in common they are 
identical co-sets. 

It is sometimes desirable to include the subgroup H among 
the co-sets as regards H. In this case, the given X sets are 
called the augmented co-sets of G as regards H. Unless the 
contrary is stated, it will be assumed that these co-sets are 
distinct, so that every substitution of G appears once and only 
once in the co-sets. The set of multiplying substitutions 
I2, . . . , t\ can be chosen in A^"' ways, h being the order of //. 

Instead of multiplying H on the right we could have mul- 
tiplied on the left. Hence all the substitutions of G can also 
be written in the form 

G=H+t'2H+ . . . -\-t\n. 

*This notation is due to Galois. 

t The concept of co-sets was used by E. Galois, but he did not use a special 
name for it. H. Weber used the term Nehengruppen to represent what we her 
call co-sets. The latter term seems not to have been used for this concepi 
before 1910, Quarterly Journal of Maihcmalks, vol. 41 (1910), p. 382. 



10) CO-SETS AND DOUBLE CO-SETS 25 

It will be proved later ( 33) that it is always possible to select 
the X 1 multipliers in such a manner that / = /, o = 2, . . ., X. 
By taking the inverses of each of the co-sets in the formula of 
the first paragraph of this section, it results that 

G^H-\-t2-'H-h . . . -\-tx-'H. 

Hence it is also possible to replace / by ta~^ in the preceding 
formula. 

If fl^i and H2 are any two subgroups of G, the symbol 

H\taH2 

is called a double co-set * of G as regards Hi and 5^2. It implies 
that each of the substitutions oi Hxta is multiplied on the right 
by every substitution of 112. All of these products are repre- 
sented by the single symbol H\tjl2. WTiile all of the substi- 
tutions of a co-set are distinct, those of a double co-set need 
not be distinct. If one substitution of G occurs exactly k times 
in such a double co-set, every substitution of the double co-set 
occurs exactly k times among the products represented by this 
double co-set. We proceed to prove this statement. 

Consider the product of the two groups H1H2, that is, all 
the products obtained by multiplying every substitution of 
Hi on the right by all the substitutions of H2. If Hi and H2 
have exactly p substitutions in common, it is clear that each 
substitution of Hi and each of H2 will appear exactly p times 
in H1H2. In fact, if h is any substitution of ^1, the p sub- 
stitutions obtained by multiplying /i on the right by the sub- 
stitutions which are comm.on to Hi and H-2 will yield the same 
products when they are multiplied on the right by Ho. As 
no other substitution of Hi can yield any of these products, 
being in the same co-set of G as regards H2, it results that 
the product H1H2 involves each one of its substitutions exactly 

* Double co-sets were first used by A. L. Cauchy, Paris Compies Rendus, 
vol. 22 (1846), p. 630. They were more fully devdopetl by Frobcnlus, 
CreUe, vol. 101 (1887), p. 273. The term double co-set was first used with this 
meaning in Bulletin of the American Mathematical Society, vol. 17 (1911), p. 292. 



26 SUBSTITUTION GROUPS (Ch. II 

p times, p being the number of the substitutions which are common 
to Hi and H2. 

It is now very easy to find the number of times that a given 
substitution of the product HitJI-z appears in this product. 
In fact, HitH2 = tta~^IIitaH2, and hence Ilitjh is the 
product of / and two groups. If a substitution s occurs exactly 
k times in ta'^Hita-Ho, the substitution t^s must occur exactly 
k times in IIitaHz. That is, each of the substitutions of H\tjl2 
occurs exactly k times in this double co-set, k being the number of 
the substitutions common to the two groups ta~^Hita and H2. 

The following special case of this theorem is often useful. 

The number of the distinct substitutions in the double co-set 
H\tjl2 is equal to the product of the orders of Hi and H2, divided 
by a number which is the common order of two subgroups contained 
in Hi and 7/2 respectively. 

The double co-set Hitjl2 is not changed if / is replaced by 
any one of the other substitutions of this double co-set, since 
the co-sets Hit^ is not altered when / is replaced by any other 
substitution of this co-set, and likewise for the co-set 4^2. 
Hence two such double co-sets of G have either no substitution 
in common or they are identical. The possible double co-sets 
of G as regards the two subgroups Hi and H2, in the given 
order, are therefore completely determined by these subgroups. 
If we count only the distinct substitutions of these double 
co-sets we may write G in the following form: 

G^Hi'H2-{-Hit2H2-\- . . -\-HityH2- 

Each substitution of G appears once and only once in the 
second member of this identity, if these symbols are used to 
represent only the distinct substitutions which occur among 
the possible products represented by the symbols. Double 
co-sets are commonly used with this restricted meaning, and 
we shall hereafter use them in this sense unless the contrary is 
stated. Hence two of these double co-sets do not necessarily 
represent the same number of substitutions, but these numbers 
are always in accord with the theorem expressed above. 



111 SYLOW'S THEOREM 27 

11. Sylow's Theorem.* The theorem of the preceding 

section may be used to prove another very fundamental theorem 
known as Sylow's theorem, which asserts that every group 
whose order is divisible by P"* but not by p"*'^^, p being a prime 
number, contains a subgroup of order />. Such a subgroup 
is called a Sylow subgroup of the group. E. Galois made the 
remark in his Manuscripts, published by Jules Tannery in 1908, 
page 39, that a group whose order is divisible by p contains 
a subgroup of order p. This theorem was proved by A. L. 
Cauchy in 1845. It was extended to the case mentioned above 
by L. Sylow, in an article published in the M athematische 
Annalen, volume 5, 1872, page 584. We shall establish this 
theorem, treating first the special case when the group G is 
symmetric and of prime power degree. 

It is evident that the symmetric group of degree p, p being 
any prime number, contains a subgroup of order p. As p\ is 
not divisible by p^ this establishes Sylow's theorem for this 
special case. We proceed to establish the theorem for the 
symmetric group G of degree p"'^^. This group contains the 
following substitution composed of p cycles: 

s = ai . . . Gp-Gp+i . . . a2p- . . . -apo+i-p+i . . . ajft+i. 

It contains also the subgroup H composed of all the substi- 
tutions on these ^"+^ letters which transform s into itself. We 
proceed to prove that H includes a Sylow subgroup of G. 

The subgroup H includes a subgroup A' composed of all the 
substitutions obtained by multiplying in every possible way 
all the substitutions generated by the separate cycles of s. 
Since each cycle is of order p and there are p"' cycles in J, it 
results that the order of i^ is 

These p" cycles are permuted under H according to the sym- 
riietric group G' of degree p" which transfonns A' into itself. 
As G' is of a smaller degree than G, and as the symmetric group 
of degree p contains a Sylow subgroup, we may assume, in a 

* A different proof of this theorem will be given in i 27. 



28 SUBSTITUTION GROUPS [Ch. II 

proof by complete induction, that G' contains a Sylow subgroup 
of order />"*', where 

m =- -. 
p-\ 

Since the p cycles of j are transformed under E according 
to a group of order p^\ and since the substitutions of U which 
permute these cycles according to these various substitutions 
transform K into itself, it results that E involves a subgroup 
E2 of order 



pn' ,^ = pm^ where m = - 



As p^ is the highest power of p which divides the order of G, 
in accord with the well-known theorem in number theory 
which has just been used, it results that G contains a Sylow 
subgroup of order />** whenever G' contains such a subgroup 
of order p^'. That is, Sylow's theorem has been established, 
by the method of complete induction, for every symmetric 
group whose degree is a power of p. 

It is now easy to establish Sylow's theorem for every pos- 
sible substitution group. Let Hi be any such group. As the 
symmetric group of degree n includes the symmetric group of 
degree n 1, it results that it is possible to find a value for a 
such that J^i is contained in the symmetric group G of degree 
pf^^. Let H2 represent any Sylow subgroup of order /?"* con- 
tained in G, and write the substitutions of G in the form of 
double co-sets as regards the two subgroups Hi and H2, as 
follows: 

G=HiH2-\-Hit2H2-\- . . . -\-H1tyH2. 

Let p^ be the highest power of p which divides the order of 
Hi. The number of distinct substitutions in each of these 
double co-sets is a multiple of />"+^-^^*, according to a theorem 
of the preceding section. This number cannot be divisible 
by p'^'^^ for every one of these double co-sets, since the order 

* This result is contained in the well-known formula for the highest power 
of a prime which divides !. Cf. Encydopidic dcs Sciences Mathimaiiques, tome 
1, vol. 3, p. 4; K. I). Carmichacl, Theory of Numbers, 1914, p. 26. 



11] SYLOW'8 THEOREM 29 

of G is not divisible by />'""'" \ Hence there must be at least 
one of these double co-sets in which 6 = /3. As />* is the order 
of a subgroup of Hi, it follows that Hi must contain a Sylow 
subgroup. Hence every possible substitution group contains 
at least one Sylow subgroup corresponding to every prime number 
which divides the order of the group. 

If Hi contains more than one subgroup of order fP, let 

^1, K2, , ^x 

represent all its subgroups of this order, A substitution s of 
Ki which is not also in K2 cannot transform A'2 into itself, 
otherwise 5 and K2 would generate a group whose order would 
be divisible by />^+^ Hence the substitutions of A'l must 
transform K2 into a complete set of conjugates under A'l and 
this set contains />" of these X subgroups. If p^' is less than 
X 1, this process can be repeated until all of these groups 
are exhausted. It is therefore necessary that X 1 be divisible 
by p. That is, the number of the Sylow subgroups of order p^ 
contained in any group is always of the form 

\+kp. 

In other words, this mmiber is always =l(mod p). 

From this theorem it results directly that every subgroup 
of order p"^ of a group G is contained in a Sylow subgroup of 
G, or is itself a Sylow subgroup of G. In fact, such a subgroup 
must transform into itself at least one of the Sylow subgroups 
of G. If it were not contained in this Sylow subgroup, G would 
involve a subgroup whose order would be a higher power of p 
than the order of its Sylow subgroup. It is also clear that 
every operator of order />* which is in G and transforms into 
itself a Sylow subgroup of order ^ must be contained in this 
Sylow subgroup. 

Another important elementary result in regard to the Sylow 
subgroups should be observed here. Suppose that the given 
X Sylow subgroups were such that they could not all be trans- 
formed into each other by the substitutions of these subgroups. 
There would therefore be a set of // which would be transformed 
only among themselves by all these substitutions. By trans- 



30 SUBSTITUTION GROUPS (Ch. II 

forming these h subgroups by the substitutions of one of their 
own number it would result that /f would be of the form \-\-k\p, 
.and by transforming the same subgroups by the substitutions 
of a subgroup of order ^ which is not included among these 
// subgroups, it would result that h would be divisible by p. 
Hence all the Sylow subgroups of order p^ in any substitution 
group constitute a single complete set of conjugates under this 
group. In fact, they constitute such a complete set under 
these Sylow subgroups. 

The three results: that Sylow subgroups of order ^ always 
exist, that their number is of the form \-\-kp, and that they form 
a single set of conjugates under the group, are very closely 
related. Generally these three results are implied by the 
expression " Sylow's theorem." All of them are of fundamental 
importance. In fact, if the theorems of group theory were 
arranged in order of their importance Sylow's theorem might 
reasonably occupy the second place coming next to Lagrange's 
theorem in such an arrangement. 

EXERCISES 

1. By means of Sylow's theorem prove that every substitution group 
of order 20 contains only one subgroup of order 5, and either one or five 
subgroups of order 4. 

2. Every group of order pq, p and q being distinct primes and P>q, 
contains only one subgroup of order p. 

3. A group of order 15 contains only one subgroup of each of the 
orders 3 and 5. Hence a group of order 15 is cyclic. 

4. Find two substitutions of order 2 such that their product is of 
order 7, and verify that they generate a substitution group of order 14. 

5. The number of subgroups of order p in the symmetric group of 
degree p is {p2) !, p being any prime number. Hence (^ 2) !s i(mod p). 

6. The number of the subgroups which are generated by cyclic sub- 
stitutions of order n and arc contained in the symmetric group of degree 

. ( 1)! , X , . , . , 

n IS , 0() bemg the totient of n. 

7. The symmetric group of degree n, n>3, contains more than one 
Sylow subgroup of order ff, whenever n! is divisible by p. 

12. Transitive Groups, and Average Number of Letters in 
its Substitutions. The two substitution groups of order 4 
1, ab'cd, ac'bd, ad'bc, 1, ab, cd, ab-cd 



121 TRANSITIVE SUBSTITUTION GROUPS 31 

represent two very important types of groups. In the former 
each letter of the group is replaced by every other letter by 
the various substitutions of the group. Such a group is said 
to be transitive. In the latter of these two groups, there is a 
letter which is not replaced by every other letter, and hence 
this group is called intransitive. Every substitution group 
evidently belongs to one and only one of these two t^-pes. Every 
symmetric group and every alternating group is transitive. 

Suppose that G is a transitive group on the n letters ai, 
a2, . . . , On. There is at least one substitution in G which 
does not involve the letter ai, viz., the identity. In general, 
the substitutions of G which omit ai constitute a subgroup 
Gi of order gi. As G is transitive it must involve a substitu- 
tion which replaces ai by a2, and this substitution transforms 
Gi into G2, G2 being composed of all the substitutions of G which 
omit 02. Hence G contains n conjugate subgroups Gi, G2, 
. . . , Gn, each of which is composed of all the substitutions 
of G which omit a letter. It is not necessary that all of these 
n subgroups be distinct. In fact, in the octic group they 
form two pairs of identical subgroups. 

All the substitutions of G can be arranged in a rectangle 
as follows, the substitutions of Gi forming the first row: 

1, S2, SS, . , Sgi 

h, S2t2, S3t2, . . ., Sg^h 
h, S2t3, 53/3, . . ., Sg^3 



t\, S2t\, Satx, . . ., Sg^t\ 



where \=g/gi. If ^2 replaces ai by 02 then all the ^i substi- 
tutions of the row involving /2 have this property. If any other 
substitution of G should replace ai by 02, all the distinct prod- 
ucts, obtained by multiplying its inverse into itself and into 
all the substitutions of the row involving /a, would transform 
02 into itself. As this would give more than gi distinct sub- 
stitutions, the row which involves /2 contains all the substitu- 
tions of G which replace ai by 02. 



32 SUBSTITUTION GROUPS [Ch. II 

Since similar remarks apply to every other row, it results 
that X = . That is, the order of tlie subgroup formed by all the 
substitutions of a transitive group which omit a given letter is 
equal to the order of the group divided by its degree. From the 
given rectangle it follows that each letter occurs gi{n l) 
times in the substitutions of G. Hence these substitutions 
involve gin{ni) =g(w 1) letters. That is, the average number 
of letters in the substitutions of a transitive group is equal to the 
degree of the group diminished by unity* In particular, the 
average number of letters in all the possible substitutions 
on n letters is n\, and this is also the average number in all 
the positive substitutions on these letters when n> 2. 

13. Intransitive Substitution Groups. One of the simplest 
examples of an intransitive substitution group may be obtained 
by multiplying together transpositions on distinct sets of let- 
ters. For instance, the intransitive group generated by the 
following three transpositions, 

ab, cd, ef, 

is of order eight and contains seven substitutions of order 2, 
besides the identity. This is a special case of the elementary 
theorem which afl&rms that h transpositions on // distinct pairs 
of letters generate a group of order 2^ and of degree 2//. This 
group is intransitive when h>\. 

By multiplying all the substitutions of any transitive group 
by all those of another transitive group, represented on a dis- 
tinct set of letters, there results an intransitive group whose 
order is the product of the orders of these two transitive groups, 
and whose degree is the sum of their degrees. Hence it is 
clear that it is possible to construct an unlimited number of 
different groups from any given group by representing the 
group on distinct sets of letters and then multiplying the sub- 
stitutions in every possible manner. Groups obtained in this 
manner are sometimes called powers of the given group, an index 
being used to indicate the number of times the group was used 

* This interesting theorem was given explicitly for the first time by G. 
Frobenius, CreUe, vol. 101 (1887), p. 287. 



13] INTRANSITIVE SUBSTITUTION GROUPS 33 

as a factor. The groups considered in the preceding para- 
graph constitute a special class of such powers. 

Another way of forming an unlimited number of non-con- 
jugate substitution groups from a given substitution group is 
by a process called establishing a (1, 1) correspondence or a 
simple isomorphism. For instance, the following three sub- 
stitution groups of order 2: 

1, ab-cd; 1, ab-cd-ef; 1, ab-cd-ef-gh 

are obtained by establishing simple isomorphisms between 
the groups 1, ab and 1, cd; I, ab, I, cd, and 1, ef; 1, aft, 1, cd, 
1, ef, and 1, gh, respectively. It is clear that we can establish 
such simple isomorphisms between any number of groups, 
obtained by writing a given transitive group on distinct sets 
of letters. All the groups thus obtained are merely different 
ways of representing the same group of order 2 considered 
abstractly, and these isomorphisms show that there is no upper 
limit to the number of letters of the substitution groups which 
represent such a group. 

The two given methods of constructing intransitive sub- 
stitution groups are called the direct prodiict method and the 
simple isomorphism method. They are the simplest methods 
for constructing such groups and the other possible methods 
are based upon them.* Before entering upon a consideration 
of other methods it should be observed that every intransitive 
group is composed of transitive constituent groups, and that it 
can be constructed by establishing some correspondence between 
these constituent groups. 

Let G be any intransitive group, and consider the letters 
which replace a given letter, a in the substitutions of this 
group. If a is replaced by b in some substitution si and there 
is also a substitution 52 in which b is replaced by c, then there 
must be a third substitution 53 = siS2 in which a is replaced by c. 
Hence a and all the letters by which a is replaced constitute 
the letters of a transitive constituent group. 

If 1, S2, . . . , 5* represent the substitutions of this con- 

* Cf. Bolza, American Journal of Mathtmatks, vol, 11 (1), p. 185. 



34 SUBSTITUTION GROUPS (Ch. II 

stituent group K, then each of these k substitutions is found 
in the same number of the substitutions of G, and hence ^ is a 
divisor of g, where g is the order of G. In the simple isomorph- 
ism method, k=g. All the substitutions of G which involve 
only the identity from K constitute an invariant subgroup 
H of G, and A' is said to be a quotient group of G as regards H. 
This quotient group is commonly represented by the following 

symbol: * 

G/H=K. 

For instance, consider the following intransitive group G: 

1, cde, ced, ab-cd, ah -de, ah-ce. 

One of the transitive constituent groups is 1, ab = K, and the 
invariant subgroup of G, which involves only the identity 
of this constituent group, is as follows: 

E = l, cde, ced. 

The quotient group G/H=K may also be regarded as a group 
in which the three substitutions of H are regarded as one sub- 
stitution, while the remaining three substitutions constitute 
the other substitution. The other transitive constituent of 
G is simply isomorphic with G, and the given correspondence 
is sometimes represented by the following symbol: 



1 




cde 


1 


ced 




cd 




de 


ab 


ce 





As an instance of a more general correspondence, we may 
consider the group of order 18 composed of all the positive 
substitutions in the direct product of the symmetric group of 
degree 3 represented on two distinct sets of letters. This 
group is represented as follows: 

This symbol was used by C. Jordan, Bulletin de la Sociili MaiMmatique de 
France, vol. 1 (1K72), p. 46. The symbol is often credited to Holder, who used 
it in 1889. Cf. H, Weber, Kleines Lehrhuch dcr Algebra, 1912, p. 192. 



14] SUBSTITUTIONS INVARIANT UNDER A GROUP 35 



1 


1 


abc 


dej 


acb 


dfe 


ah 


de 


he 


ef 


ac 


df 



If we let K represent the former of these two constituent 
groups it is clear that H=l, def, dfe, and that G/n=K can be 
regarded as a group in which three substitutions of G are con- 
sidered as a single substitution. 

14. Substitutions which are Commutative with Each of the 
Substitutions of a Transitive Group. When Gi, the subgroup 
composed of all the substitutions which omit a given letter, 
reduces to the identity, the transitive group G is said to be a 
regular group. Regular groups are especially important since 
every possible group of finite order can be represented as a 
regular substitution group, as will be proved later. We pro- 
ceed to prove the important theorem, which we shall call 
Jordan's theorem* that the total number of substitutions on 
n letters which are commutative with every substitution of a 
regular group G on the same n letters constitutes a group C 
which is conjugate or similar to G; i.e., it can be transformed 
into G by some substitution. This theorem results ahnost 
immediately if we multiply all the substitutions of (7=1, 
^2, S3, . . ' , Sg successively on the right and on the left as 
follows: t 



G = 



* This theorem was first proved in Jordan's thesis, Paris, 1860, p. 39. 

t The two simply isomorphic regular substitution groups which are repre- 
sented by these square arrays have been called potential and antipotential groups 
respectively. G. FrattinI, Aui della R. Accademia del Uncei. Memoria, vol. 14 
(1883), p. 144. 



1, 


S2, S3, 


. . ., Sg 




1, 


S2, 


S3, . 


. ., Sg 


^2, 


S2^, S3S2, 


., SfSo 




^2, 


52^ 


52^3, . 


. ., SiSg 


^3, 


S2S3, S3^, 


. . ., SgS3 


G' = 


^3, 

Sg, 


^3^2, 
S9S2, 


^3^ . 

V3, . 


. ., S3Sg 


Sg, 


S2Sg, SaSg, 


r 2 
. ., Sg 


. ., V' 



36 SUBSTITUTION GROUPS [Ch. II 

The permutation of the substitutions of each of these rows 
as regards the first row represents a substitution, and we may 
assume, without loss of generality, that the substitutions repre- 
sented by the first square, when the various rows are associated 
successively with the first row, are the substitutions of G. Any 
one of these rows, say the one involving Sa, represents a substi- 
tution obtained by multiplying all the substitutions of G on the 
right by Sa', while any row of G', say the one involving s^, 
represents a substitution obtained by multiplying all the sub- 
stitutions of G on the left by Sp. Since we get the same result 
when we multiply all the substitutions of G first on the right by 
Sa and then on the left by Sp, as when we multiply them first 
on the left by Sp and then on the right by Sa, as a consequence of 
the associative law, it follows that each substitution of G is 
commutative with every substitution of G', and vice versa. 
Moreover, G' includes all the substitutions on these letters, 
which are commutative with every substitution of G, since 
every such substitution must involve all the letters of G and 
the totality of these substitutions forms a group. 

As G and G' are different ways of representing the group 
G they must be simply isomorphic. It remains to prove that 
they are conjugate. If we establish a simple isomorphism 
between G and G' in such a way that all the substitutions begin 
with the same letter, the second letters in all the substitutions 
of these groups represent the substitution by means of which 
G may be transformed so that the first two letters in each 
one of its substitutions are the same as the first two letters 
of the corresponding substitution in G'. Since this transfor- 
mation will lead to simply isomorphic groups, and since two 
simply isomorphic regular groups have the property that the 
corresponding substitutions are identical whenever the first 
two letters of all these substitutions are the same, we have 
proved Jordan's theorem; viz., with every regular group of 
order n there is associated another regular group of order n such 
tltat each of tliese groups is composed of the total number of substi- 
tutions on these n letters which are commutative with every substi- 
tution of the other group. 



14] SUBSTITUTIONS INVARIANT UNDER A GROUP 37 

The two groups G and G' whicli are defined by Jordan's 
theorem are called conjoints. When G is abelian it coincides 
with its conjoint and vice versa. A special case of this theorem 
relating to a cyclic group was proved above in 8. Suppose 
that G is transitive and of degree n, but not regular, and that 
all the substitutions of G which omit ai omit also a2, . . . , 
Qa- Hence Gi, which is composed of all the substitutions of G 
which omit ai, is transformed into itself by all the substitutions 
of G which replace ai by a2, . . . , That is, Gi is trans- 
formed into itself by a subgroup H of order agi. All of the 
substitutions of H, except those of Gi, must involve each of 
the letters ai, a2, . . . , da- Hence H is an intransitive group, 
and the components of its substitutions involving the letters 
ai, 02, . . . , Oa must form a regular group. This regular 
group is a transitive constituent of H and the subgroup Gi, 
which is transformed into itself by H, corresponds to the iden- 
tity of this transitive constituent. Let Ci be the conjoint 
of this constituent, and consider the n/a transforms of Ci 
under G. 

A (1, 1) correspondence can be established between the 
substitutions of these n/a transforms such that each substitu- 
tion is of degree n and is transformed into itself by G. To do 
this it is only necessary to regard as one substitution all the 
transforms of a single substitution of Ci. This proves a theorem 
due to H. W. Kuhn,* which is a generalization of Jordan's 
theorem and may be stated as follows: 

A necessary and sufficient condition that there are a substi- 
tutions on the letters of a transitive group, which are commuta- 
tive with every substitution of the group, is that its subgroup which 
is composed of all its substitutions which omit one letter omits 
exactly a letters. 

If Gi is of degree 1, the identity is therefore the only 
substitution on these n letters which is commutative with 
every substitution of G, as is also otherwise evident. 

* American Journal of Mathematics, vol. 26 (1904), p. 67. 



38 SUBSTITUTION GROUPS [Ch. 11 

EXERCISES 

1. Every transitive group of degree n involves at least n 1 substitu- 
tions which separately involve all the letters of the group; if it contains 
also substitutions of degree na, !<<, it must contain more than 
1 substitutions of degree n. 

Suggestion. The average number of letters in the substitutions is n 1. 

2. If all the substitutions which omit a given letter of a transitive 
group of degree n constitute a group of degree nl, then the n conjugates 
of the latter are transformed under the transitive group in exactly the same 
way as its letters are transformed. 

3. A transitive group composed of invariant substitutions is necessarily 
regular. 

4. If the order of a transitive group is p^, p being a prime number, 
the subgroup composed of all its substitutions which omit a given letter 
omits />"* letters, where mo>0. 

5. All the substitutions of highest degree (n) in any transitive group 
generate a transitive group of degree w, which either coincides with the 
original group or difTers from it merely with regard to substitutions of 
degree 1. 

Suggestion. Use the theorem that the average number of letters 
in all the substitutions of a transitive group of degree n is nl, while in 
an intransitive group this number is smaller. 

6. The total number of the substitutions which are conjugate to a given 
substitution of a group must generate either the entire group or an 
invariant subgroup. This theorem remains true if the word conjugate 
is replaced by the word similar. 

7. The number of the substitutions of degree p^ and of order p in the 
symmetric group of degree n is prime to p whenever p^ is the highest power 
of P which does not exceed n. Cf. Annals of Mathematics, vol. 16 (1915), 
p. 169. 

15. Primitive and Imprimitive Groups. When the sub- 
group Gi, composed of all the substitutions of a transitive 
group G which omit a given letter, is of degree na, n being 
the degree of G, we have seen that the letters of G may be 
divided into n/a sets, each set involving a distinct letters, such 
that these sets are transformed as units by all the substitutions 
of G. Such a substitution group is said to be imprimitive 
whenever o> 1, and the sets of a letters are called its systems 
oj imprimitivily. These systems of imprimitivity are trans- 
formed under G according to a transitive group which has a 



15] PRIMITIVE AND IMPRIMITIVE GROUPS 39 

(1, w) correspondence with G. If m> 1, G must therefore con- 
tain an invariant subgroup of order m which transforms each 
of these systems of imprimitivity into itself. If m = l, each 
substitution of G, besides the identity, must transform at least 
one of these systems of imprimitivity into another. 

While every transitive group whose Gi omits more than one 
letter is necessarily imprimitive, it does not follow that the Gi 
of an imprimitive group must omit more than one letter. We 
proceed to prove that a necessary and sufficient condition that G 
is imprimitive is that Gi is contained in a larger subgroup of G. 
In other words, a necessary and sufficient condition that G is 
imprimitive is that Gi is a non-maximal subgroup of G. It 
should be observed that this theorem connects the theory of 
imprimitivity with the theory of abstract groups. Every 
transitive substitution group which is not imprimitive is said 
to be primitive. 

The given theorem is contained in a more general theorem 
which may be stated as follows: A necessary and sufficient 
condition that a complete set of conjugate substitutions or sub- 
groups of G is transformed under G according to an imprimitive 
substitution group is that the largest subgroup of G which trans- 
forms into itself one of these substitutions or subgroups is contained 
in a larger subgroup of G. That this condition is sufficient 
results from the fact that if this largest subgroup K, which 
transforms into itself a substitution or subgroup L, is contained 
in a larger subgroup H, then the number of the conjugates of 
L under H is equal to the quotient obtained by dividing the 
order of H by the order of K, and H involves all the substi- 
tutions of G which transform these conjugates among them- 
selves. 

Every substitution of G which is not in 77 must therefore 
transform the set of conjugates of L under // into an entirely 
new set of conjugates, and therefore this set of conjugates is 
transformed as a unit. Hence all the conjugates of L can be 
divided into sets such that they are all transformed as units 
and such that no two sets have a common substitution or sub- 
group. On the other hand, if the complete set of conjugates 



40 SUBSTITUTION GROUPS [Ch. li 

to which L belongs is transformed according to an imprimitive 
group there must be such a group as H which includes A', since 
all the substitutions which transform among themselves the 
substitutions or subgroups, corresponding to a system of im- 
primitivity, must constitute a group. Hence the given general 
theorem is established. 

To deduce the special case relating to the primitivity of 
G when Gi is of degree w 1, we have only to observe that a 
necessary and sufficient condition that G\ is transformed into 
itself by only its own substitutions, is that its degree is w 1. 
As G transforms the conjugates of G\ in exactly the same way 
as it transforms its letters, whenever the degree of Gi is w 1, 
and as we assume in the present case that the degree of Gi is 
exactly 1, it results directly from the given theorem that G 
transforms the conjugates of Gi, and hence also its letters, 
according to a primitive group^ whenever Gi is maximal, and 
only then. 

If G\ is a transitive group on 1 letters, G is said to be 
doubly transitive, and every pair of letters of G is transformed 
into every other pair by the substitutions of G. In general, 
G is said to be r-Jold transitive, whenever Gi is (r l)-fold trans- 
itive and of degree w 1. Since the order of a transitive group 
is always a multiple of its degree, it results that the order of 
an r-fold transitive group is a multiple of n{n\) . . . 
(n r+l). A group which is more than simply transitive 
is said to be multiply transitive. The alternating group of 
degree n is ( 2)-fold transitive and the symmetric group of 
degree n is said to be either -fold or ( l)-fold transitive. We 
shall generally say that this group is (w l)-fold transitive. 
With the exception of the alternating and the symmetric groups 
no group is known which is more than five-fold transitive and 
only two such five-fold transitive groups are known. These 
groups are of degrees 12 and 24 respectively and were dis- 
covered by E. Mathieu in 1861. The theory of multiply 
transitive groups has not yet been extensively developed, and 
it seems to offer great difficulties. 



161 GKOUFS ON FOUR LE1TEU8 41 

EXERCISES 

1. If all the substitutions of order w, m>2, in a group are conjugate, 
they must be transformed under the group according to an imprimitive 
group. 

Suggestion: The generating substitutions of one cyclic subgroup are 
transformed into all of those of another. 

2. If an imprimitive group contains substitutions besides the identity 
which do not interchange any of its systems of imprimitivity, in a given 
set of systems of imprimitivity, all such substitutions constitute an invari- 
ant subgroup. 

3. Every regular group of composite order is imprimitive, and involves 
as many different sets of systems as it has subgroups, excluding the identity. 

4. A transitive group of order p", p being a prime number and o>l, 
is always imprimitive. 

Suggestion: Consider its complete sets of congjuate substitutions 
and observe that each of these sets involves p^ distinct substitutions. 

5. Every invariant subgroup besides the identity of a primitive group 
is transitive. 

6. The total number of substitutions which are commutative with 
every substitution of the intransitive group obtained by establishing a 
simple isomorphism between n, n>2, symmetric groups of degree n, 
written in distinct sets of letters, constitute a conjugate intransitive group. 

16. Groups Involving no More than Four Letters. The 

only possible substitution group on two letters is 1, ah. Since 
every system of intransitivity must involve at least two letters, 
a group of degree 3 is necessarily transitive. It is also included 
in {abc)all* and hence its order is a divisor of 6. Therefore 
{ahc)all and {ahc) are the only two possible groups of degree 3. 
If a group of degree 4 is intransitive, each of its two systems of 
intransitivity must be of degree 2. The largest intransitive 
group of degree 4 is therefore the direct product of {ah), (cd). 
The other possible intransitive group is a simple isomorphism 
between the substitutions of these transitive groups of degree 2. 
Hence there are two and only two intransitive groups of degree 4; 

* The notation (aioj . . . an)aU is used to represent the symmetric group of 
degree n, while (aiOj . . . On) represents the group generated by the cyclic sub- 
stitution fliOj ... On. It should, however, be emphasized that the symbol 
(aiOi ... On) is also often employed to denote the substitution diai . . . ii,. 
Since there is no uniformity of usage along this line, the reader is frequently 
obliged to determine the meaning from the context. 



42 SUBSTITUTION GROUPS [Ch. II 

one of these is of order 4 while the other is of order 2. Their 
substitutions are as follows: 

1, ab, cd, ab-cd; 1, ab-cd. 

According to Sylow's theorem, (abed) all involves at least 
one subgroup of order 8 and all its subgroups of this order are 
conjugate. Since all conjugate groups are regarded as identical 
in the enumeration of groups, there is one and only one sub- 
stitution group of degree 4 and of order 8. This is known 
to be transitive (2). We shall represent it by the symbol 
{abcd)B. Since the order of every transitive group is a mul- 
tiple of its degree, and since a group must be included in the 
symmetric group of its own degree, it results that the order 
of a transitive group of degree four is 4, 8, 12 or 24. 

As there is one and only one such group of each of the orders 
8 and 24, it remains to determine all the possible groups of 
orders 4 and 12. We know that there are two transitive groups 
of the former order; viz., the subgroups of the octic group con- 
sidered in 2, and there is one group of the latter order; viz., 
{abcd)pos* We proceed to prove that no other groups of these 
orders are possible. Another transitive group of order 4 would 
also be regular, since the average number of letters in its sub- 
stitutions is 3. It could not be cyclic, since there is one and only 
one cyclic group of each order, and two simply isomorphic 
regular groups are conjugate. If it were non-cyclic it would 
involve all the possible substitutions of the form ab-cd in the 
symmetric group of degree 4. This proves that there are only 
two regular groups of degree 4. One of these has three con- 
jugates under the symmetric group while the other is invariant 
under this group. 

To prove that there is only one group of order 12 and degree 
4, we observe that every such group would have to contain 
a subgroup of order 3 according to Sylow's theorem. As all 
the subgroups of order 3 in {abcd)aU are conjugate, we may 
assume that every group of order 12 and degree 4 includes 
iflbc). Since this subgroup could not be invariant under a tran- 

* The symbol (aioj . . . an)pos represents the alternating group of degree n. 



17] SIMPLICITY OF THE ALTERNATING GROUP 43 

sitive group of degree 4, every group of this degree and of order 
12 must involve the 4 subgroups of order 3 in {abcd)all, and 
hence it must be identical with {ahcd)pos. That is, there are 
exactly five transitive groups of degree 4, one of each of the orders 
24, 12, 8, and two of irder 4. Each of these possible transitive 
groups has at least two substitutions in common with the non- 
cyclic regular group. 

It has been observed that (abcd)pos cannot involve a sub- 
group of order 6, since its Sylow subgroups of order 3 are trans- 
formed into themselves by their own substitutions only under 
{abcd)pos. We have here an instance where a group whose 
order is divisible by 6 does not contain a group of order 6. 
That is, while the order of every subgroup is a divisor of the 
order of the group there is not neccessarily a subgroup for each 
divisor of the order of a group. The ten groups whose degrees 
do not exceed 4 may be replresented as follows: * 

(a6), {abc)all, (abc), {abcd)all, {abcd)pos, 
{abed) 8, (abed), (abed) 4, iab)(cd), (ab-cd). 

17. Simplicity of the Alternating Group of Degree n, n?^4. 

If a group does not contain any invariant subgroup besides 
the identity it is said to be a simple group. All other groups 
are said to be composite. We proceed to prove that the alter- 
nating group is simple except when its degree is 4. Suppose 
that si, 52 are two cyclic substitutions of order greater than 2, 
such that one can be obtained from the other by interchanging 
two adjacent letters. That is, 

si= ... ayOaOfi . . . , 52= . . . ayOfiaa . . . 

Hence SiS2~^=at^yap. That is, if the order of a substitution 
exceeds 2 it is always possible to find anotJier substitution similar 
to it such that the product of the two is a cycle of order 3. By 
means of this elementary theorem and the fact that every posi- 

* A fundamental problem of substitution groups is the determination of all 
the substitution groups of degree w. This problem has been completely solved 
when n does not exceed 11. The noted French mathemalidan .\. L. Couchy 
was the first to do serious work along this line; Paris Comptes Rendus, vol. 21 
(184.5), p. 136a. 



44 SUBSTITUTION GROUPS [Ch. II 

tive substitution is the product of cycles of order 3, it is easy 
to prove that every alternating group whose degree exceeds 
4 is simple. 

Assume that G is the alternating group of degree , n>4, 
and that it involves an invariant subgroup H. As H involves 
all the substitutions of G, which are conjugate with any one 
of its substitutions under G, it cannot involve any substitutions 
of order 2 without also containing substitutions whose orders 
exceed 2. This results immediately from the fact that the 
positive substitution ah-cd . . . of order 2 is transformed by 
hde into a substitution which is not commutative with it, and 
hence the product of these two substitutions has an order which 
exceeds 2. If H is not the identity, it must therefore involve 
either substitutions of an odd prime order, or substitutions 
involving cycles of even order greater than 2. As the cyclic 
groups generated by such "feubstitutions are evidently transformed 
into themselves by negative substitutions, these cyclic groups 
are transformed into all their conjugates under the symmetric 
group of degree n by substitutions of G. That is, H must 
involve all these conjugates, and hence it must involve all 
the substitutions of the form abc in G. In other words, H 
coincides with G whenever it exceeds the identity. This proves 
the important theorem: the alternating group of degree n is 
simple whenever w>4. 

The alternating group of degree 3 is evidently simple, but 
the alternating group of degree 4 contains 1, ah-cd, ac-hd, 
ad-bc a.s an invariant subgroup, as was observed above. Hence 
every alternating group, with the exception of the alternating 
group of degree 4, is simple. In this special case the alternating 
group contains a group of order 4 as an invariant subgroup. 
From what precedes, it results immediately that the s>Tnmetric 
group of degree w contains only one subgroup of order !/2. 
If it contained two such subgroups, one would contain only 
!/4 positive substitutions, and these would constitute a sub- 
group of the alternating group. This is impossible, since such 
a subgroup would contain all the substitutions of odd order and 
hence all the substitutions of the form abc, 



18] GROUPS OF DEGREE FIVE 46 

18. Groups of Degree Five. The intransitive groups on 
five letters must involve transitive constituents of degrees 
3 and 2, and hence all of them are contained in the direct 
product of {abc)all, (de). Consequently there are three such 
groups one of order 12 and two of order 6. They may be 
represented as follows: 

{abc)all(de), {abc)(de), \{abc)all(de)\pos. 

AU the transitive groups of degree 5 are primitive, and they 
involve either only one, or six subgroups of order 5. As these 
six groups must generate the alternating group, according to 
the preceding section, it follows that the alternating and the 
symmetric groups of degree 5 are the only ones which involve 
six subgroups of order 5. If such a group contains only one 
subgroup of order 5, its order divides 20, since a generating 
substitution of the subgroup of order 5 is transformed into 
itseK by only five substitutions on these 5 letters. We may 
evidently suppose that each of the possible groups involves 
the same subgroup of order 5 since all these subgroups are con- 
jugate. Hence there is only one such group of each of the orders 
20, 10, 5. The five transitive groups of degree 5 may be repre- 
sented as follows: 

(abcde), (abcde) 10, (abcde)20, (abode) pos, {abcde)all. 

The group (abcde) pos is especially interesting since it is 
the simple group which has the smallest possible comixjsite 
order. This group is simply isomorphic with the total number 
of movements which transform the icosahedron into itself, as 
we shall see later, and hence it is frequently called the icosa- 
hedron group. Galois was the first to observe that it is simple, 
and he also observed that it is the smallest simple group of 
composite order. Sir W. R. Hamilton observed, in 1856, that 
this group may be defined abstractly as the group generated 
by two substitutions of orders 2 and 3 respectively whose 
product is of order 5. That is, every pair of substitutions 
which satisfy these conditions generates a group of order 00, 
which is simply isomoqihic with (abcde)pos. 



46 SUBSTITUTION GROUPS [Ch. II 

19. Holomorph of a Regular Group. If G is a regular group 
of order n, all the substitutions on these n letters which trans- 
form G into itself constitute a group which has been called the 
holomorph * of G. For instance, the symmetric group of 
degree 3 is the holomorph of its subgroup of order 3, {ahcd)^ 
is the holomorph of {abed), and {abcd)20 is the holomorph of 
{abode). The holomorph of G includes the conjoint of G, and 
hence it is also the holomorph of this conjoint. If this conjoint 
is not identical with G, it is conjugate with G under a substitu- 
tion of order 2 which transforms the holomorph of G into itself. 
This substitution and G generate a group known as the double 
holomorph of G. Every non-abelian group has a double 
holomorph. 

Since the holomorph K ol G involves exactly n substitutions 
which are commutative with every substitution of G, it must 
transform the substitutions of G in k/n different ways, k being 
the order of K. The largest subgroup of degree n\ con- 
tained in K must also transform the substitutions of G in k/n 
different ways. This subgroup is known as the group oj 
isomorphisms of G.f Hence the order of the holomorph of a 
group is the product of the order of the group and the order of its 
group of isomorphisms. Since any two simply isomorphic 
regular groups are conjugate, the group of isomorphisms of G 
transforms G into every possible simple isomorphism with itself. 
That is, it transforms it into all its possible automorphisms. 

If a group involves substitutions which transform it into 
every possible automorphism, but does not contain any invari- 
ant substitution besides the identity, it is said to be a complete 
group. The symmetric group of degree 3 is evidently a com- 
plete group. The holomorph of a complete group is the product 
of the group and its conjoint. It is easy to prove that the 

The concept of holomorph was used by many early writers, but the term 
was introduced by W. Burnside in the first edition of his Theory of Groups, 
1897, p. 228. 

t The statement relating to this matter in the EncyklopUdie der Mathemalischen 
Wissenschaften, vol. 1, p. 221, note 103, is inaccurate. The group of isomorphisms 
is one of the most important and also one of the most far-reaching concepts in 
group theory. 



20] CLASS OF A SUBSTITUTION GROUP 47 

symmetric group of degree 4 is also complete. In fact, this 
group is generated by any two of its cyclic subgroups of order 
4. One of the generators of such a cyclic group can be selected 
in six different ways, and after it has been selected, a generator 
of the second cyclic group can be selected in four different ways. 
The two generators can therefore be selected in 24 different 
ways. Since (abcd)all contains no invariant substitution 
besides the identity, it must transform its own substitutions 
in these 24 possible different ways. Hence the holomorph of 
(abed) all is the direct product of this group and its conjoint. 
The order of this holomorph is therefore 576. 

EXERCISES 

1. The symmetric group of degree n does not contain any subgroup 
of index p, if p is greater than 2 but less than the largest prime factor of n. 

(Cauchy, 1815.) 
Suggestion: Such a subgroup could not involve all the substitutions 
whose order is this prime factor, since these substitutions would generate 
the alternating group. 

2. Prove that the symmetric group of degree n is generated by a 
cyclic substitution of degree n1 and a transposition which connects 
any one of these letters with the remaining letter. 

3. The group of order 48 on Xi, . . ., Xi, which transforms the func- 
tion XiXi+xsXt+XiXi into itself is imprimitive and involves an invariant 
subgroup of order 8. 

4. The polynomial {xi+(t)Xi+ 0)^X3+ . . . +c/~^XrY is transformed 
into itself by the cychc group of order r on these variables, if w represents 
a number whose powers give all the rth roots of unity. 

Suggestion: If we multiply the polynomial within the parenthesis, 
which is known as the Lagrangian resolvent, by a power of u, we inter- 
change the variables cyclically. 

5. Construct all the possible substitution groups that can be repre- 
sented on six letters. Cf. American Journal of Mathematics, vol. 21 (1899), 
page 327. 

20. Class of a Substitution Group. A substitution group 
G of degree n is said to be of class na, a<n, if it contains at 
least one substitution of degree na but does not contain 
any substitutions besides the identity whose degree is less than 
n-a. For instance, every symmetric group is of class 2 while 
every alternating group is of class 3. It is easy to prove that 



48 SUBSTITUTION GROUPS (Ch. II 

these two infinite systems are composed of all the possible 
primitive groups which are of class 2 and class 3 respectively. 
A substitution which actually contains exactly k letters is some- 
times said to be of class k. 

To prove that a primitive group G which is of class 2 and 
of degree n is symmetric, it is only necessary to observe that 
such a group contains at least two transpositions having a letter 
in common, otherwise G would be imprimitive. Hence G 
involves the symmetric group of degree 3. It must therefore 
contain two such symmetric groups which have two letters 
in common. Hence G contains the symmetric group of degree 
4. By continuing this process, it results that G is the symmetric 
group of degree n. In exactly the same manner, it can be proved 
that if a primitive group of degree n contains a substitution of 
the form ahc it includes the alternating group of degree n. That 
is, if a primitive substitution group involves a transposition it is 
a symmetric group, and if it involves a substitution of the form abc 
without also involving a transposition, it is an alternating group. 

Suppose that G is a primitive group which contains a sub- 
stitution of degree and of order p, p>S, and that G is of 
degree n, n>p, p being a prime number. There must be at 
least two substitutions in G which are of degree and of order p, 
and which have some but not all their letters in common. Let 
5i and 52 be two such substitutions. If S2 involves more than 
one letter which is not also contained in si, there is some 
power of Si in which two such letters are adjacent. The 
transform of S\ by this power will then be a substitution which 
has more letters in common with s\ than 52 has; but this trans- 
form involves at least one letter which is not contained in si. 
Hence we may assume that G contains two substitutions of 
degree p and of order p such that these substitutions contain 
exactly p\ common letters. These two substitutions generate 
a doubly transitive group of degree p-\-\. By continuing these 
considerations it results that if a primitive group of degree n 
contains a substitution of degree p and of order p, p being any 
prime number, this primitive group is at least {n p-{-\)-fold 
transitive. 



20] CLASS OF A SUBSTITUTION GROUP 49 

It is now easy to prove that a primitive group of class ^, 
/>>3, cannot have a degree which exceeds />+2. In fact, 
if such a group were of degree />+3, it would be at least four- 
fold, or four times, transitive. Since any set of four letters 
of such a group can be replaced by an arbitrary set of four let- 
ters, it results that any four times transitive group must involve 
an intransitive subgroup H which has the sjTnmetric group of 
degree 3 for one constituent, and a transitive group on the 
remaining letters for the other constituent. In the present 
case the latter group is of degree p. 

In any transitive group of degree />, the subgroups of order 
p generate a simple group, since an invariant subgroup of a 
primitive group is transitive. If this simple group is not the 
entire group, it must be invariant under the entire group and 
the corresponding quotient group must be cyclic, since it is a 
subgroup of the group of isomorphisms of a group of order p, 
and this group of isomorphisms is cyclic, since p has primitive 
roots. Hence it results that H includes substitutions of the 
form ahc whenever G is of degree />+3, since its transitive con- 
stituent of degree p cannot give rise to a quotient group which 
is simply isomorphic with the symmetric group of degree 3 
when />>3. That is, if a primitive group is of class p, p being 
a prime number greater than 3, the degree of this primitive group 
is at most p-\-2. 

There is evidently one and only one primitive group of 
degree p and of class />; viz., the group of order p. In order 
that a primitive group of degree />-fl be of class p, it is clearly 
necessary that this group be of order />(/>+l), and that it 
contain (/>+!)(/> 1) substitutions of order />. Hence p-\-\ 
must be of the form 2*", and if. p-\-\ is of this form there is one 
and only one such group. A primitive group of degree p-\-2 
which is of class p must therefore include this primitive group 
of degree />+l. It must also contain a substitution of order 
2 and of degree p-\-\ which transforms into its inverse one of 
the substitutions of order p in this group of degree />+ 1 . Hence 
there cannot be more than one primitive group of degree 
p-\-2 and of class />. 



60 SUBSTITUTION GROUPS [Ch. II 

The fact that this primitive group actually exists was proved 
by C. Jordan,* but we shall not give the proof here. We have, 
however, established the following theorem: 

When the prime number p, p>S, is not of the form 2'*! 
there is one and only one primitive group of class p. When p is 
of the form 2* 1 there cannot be more than three primitive groups 
of class p. 

The study of primitive substitution groups by means of 
their classes was begun by C. Jordan, who proved that there is 
a finite number of such groups of every class. In recent years 
W. A. Manning has contributed new theorems on this interest- 
ing but difficult subject, t 

EXERCISES 

1. If a primitive substitution group contains two transitive subgroups 
which can be transformed into each other by a transposition, the primitive 
group is alternating or symmetric. 

2. Prove that the two substitutions abe cdf a.nd eg fc/ generate a group 
of order 168, and that this group is simple. 

3. It is known that the simple group of order 504 can be represented 
as a transitive substitution group on 9 letters. Hence prove that it con- 
tains the abelian group of order 8 which is composed of seven substitu- 
tions of order 2 and the identity. 

* Journal de Mathtmatiques (2), vol. 17 (1872), p. 3.51. 

t W. A. Manning, Transactions of the American Mathemalical Society, vol. 
11 (1912), p. 375. 



CHAPTER III 

FUNDAMENTAL DEFINITIONS AND THEOREMS OF 
ABSTRACT GROUPS 

21. Introduction. In the preceding chapters we gave 
several examples of groups which were either in the form of 
substitution groups or could readily be associated with such 
groups. Some of the fundamental theorems of substitution 
groups were also developed. As the theory of groups is applic- 
able to many different subjects, it became a matter of economy 
of thought to develop and to state the main theorems of this 
theory in a language which is common to these various sub- 
jects. The efforts to accompHsh this end led gradually to 
what is known as abstract group theory. 

Many of the theorems proved in the preceding chapter 
can be used directly in the theory of abstract groups. In fact, 
we shall find that the theory of substitutions is a very useful 
means to study the abstract properties of groups. It has 
already been observed, 15, that the question of primitivity and 
imprimitivity of a substitution group has an abstract meaning, 
even if these terms apparently relate only to the notation of 
substitution groups. 

Some of the definitions used in the theory of substitution 
groups are not directly available for the study of abstract 
groups, and hence it will Be necessary to re-define some of the 
terms used in the preceding chapter. This is due to the fact 
that substitutions have inherent properties. For instance, 
the associative law is always satisfied when substitutions are 
multiplied, and hence we did not need to specify this law in 
defining a substitution group. On the contrary, this law 
should be included in a definition of an abstract group. 

51 



52 ABSTRACT GROUPS (Ch. Ill 

22. Definition of an Abstract Group and a Few Properties 

of its Elements. In the theory of finite abstract groups we 
deal with a set of distinct symbols G=si, S2, . . . , Sg, and we 
assume that any two of them can be combined according to 
some law which is called multiplication, and which is denoted 
in the same way as multiplication is commonly denoted. This 
set of symbols represents a group provided the symbols satisfy 
the following conditions: 

1. If any two of the three symbols in an equation of the form 

are contained in G, then the third is also contained in G, and it 
is completely determined by this equation. It is assumed that 
this statement includes the case when the two symbols which 
are contained in G are identically equal to each other. 

2. The symbols of G obey the associative law. That is, 

iSaSfi)Sy=S{SpSy). 

From the former of these conditions it results that if one 
of the three s3niibols in 

SaSfi = Sy 

remains fixed while another assumes successively all the values 
of the symbols or elements of G, the third will also run through 
all these elements. In particular, there must be an element 
si such that 

SiSp=Sfi. 

Such an element is called the left-hand identity of Sp. This 
left-hand identity is the same for all the elements of G. To see 
this fact we multiply the last equation on the right by Sy, and 
then let Sy run through all the elements of G. In exactly the 
same way it may be observed that G contains only one right- 
hand identity s[. 

To prove that"5l=5i it may be observed that if we replace 
S0 by s[ in the last equation it results that 

SiSi = Si. 



221 DEFINITION OF ABSTRACT GROUPS 53 

Similarly, by letting Sa = si in the following equation 

it results that 

Sl5i=Si. 

That is, 5i = Sii in other words, every group contains one and 
only one element which does not alter the value of any element 
when it is used as a multiplier on the right or on the left. This 
element is called the identity of the group, and it is denoted by the 
symbol 1. Every other element changes all the elements into 
which or by which it is multiplied. 

Since g is finite there must be some finite power of ^^ (a = l, 
2, . . , g) which is equal to some other power of this element. 
That is, the series 

'a; ''a J 'a } j 'a 

must involve a term which is equal to a preceding term when 
n is taken sufficiently large. Let sj be the lowest power of s^ 
which satisfies the equation 

sj = sj'\ or sj'^^ = sj, 0<k<r. 
Since 

and since G involves only one identity, it results that sj=l= s^. 
That is, the lowest power of Sa which is equal to a lower power 
occurs after the identity appeared in the series of successive 
positive powers. On the other hand, since 



it results that r = ^+l. That is, the first power which is equal 
to a lower power in a series of successive powers of an clement 
is the one following the identity in this series. Hence the series 
is periodic and the number of different elements in each period 
is k. This number is called the order or period of s. 

Two powers of Sa whose exponents are of the form l+mk 
are equivalent whenever m is any positive integer or zero. 
Negative exponents are introduced by assuming that this 

* The symbol Ja" indicates the product ia-ia- 5a . . takenw times as factor; 
Sa is defined as equal to the identity. 



54 ABSTRACT GROUPS (Ch. Ill 

equivalence remains true when w is any negative integer. In 
particular, two elements of G which satisfy the equation 

SaSfi = 1 

are said to be the inverses of each others and S0 = Sa~^ is denoted 
by Sa~^. Elements of order 2 are their own inverses; but all 
other elements, besides the identity, go in pairs, composed 
of an element and its inverse. In particular, every possible 
group contains an even number of elements, which may be zero, 
of every order which exceeds 2. 
Since 

SaSp . . , Sx-Sx~^ . . . Sp-'^S-^ = 1, 

it results that the iilverse of SaSp . . . s\ is Sx~^ . . . Sp~^Sa~\ 
and that 

If all of the elements Sa, Sp, . . . , Sx are of order 2, then 

(SaSfi . . . Sx)-^=Sx . . . SpSa. 

In particular, if the product of two elements, Sa, Sp of order 2 is 
also of order 2 the elements are commutative, that is SaSfi = S0Sa. 

If the elements Sa,- sj^, . . . , 5^* = 1 do not include all the 
elements of G, they represent a subgroup of G. By exactly the 
same arguments as were used to prove that the order of a sub- 
stitution group is divisible by the order of each of its subgroups 
(9), it can be proved that the order of an abstract group is 
divisible by the order of each one of its subgroups. Hence it 
results that ^ is a divisor of g. 

23. The Cyclic Group.* By definition the cyclic group 
is generated by a single element, and every group which can be 
generated by a single element is cyclic. If the order of such 
a group is g, it contains at least one element 5 of the order g. 
The ordel" of s*, m being an arbitrary positive integer, is g/d, 
d being the greatest common divisor of m and g. If <l>{g) reprc- 

Many of the results of this section can be deduced from properties of the 
n roots of unity. In fact, these n roots form a cyclic group with respect to multi- 
plication. 



23] THE CYCLIC GROUP 55 

cnts, as usual, the number of natural numbers prime to g 
and not greater than g, then G involves exactly <l>(g) elements 
of order g, and hence it involves <f>(g) generators. All cyclic 
groups of the same order are simply isomorphic, and hence we 
shall say there is one and only one Cyclic group whose order is 
an arbitrary natural number. This group of order g may be 
represented by the ggth. roots of unity, when they are combined 
by multipUcation. 

If m is prime to g, then .5'" generates s; that is, every element 
is generated by any one of its powers whose index is prime to the 
order of the element. This theorem is included in the statement 
that if J, is any number such that the highest common factor of 
g and k is d, then will s^ generate the cyclic subgroup of order , 
g/d, which is also generated by s*^. It should be observed that 
a necessary and sufficient condition that two elements of G 
generate each other is that they have the same order, and a 
necessary and sufficient condition that one of these elements 
generates the other is that the order of the former is divisible 
by that of the latter. /"/ 

Let./ be an elemen{ of any group such that f^ is of order n. 
If all the prime factors of m are also in n. it results from this 
that the order of / is equal to mn. In general, the given con- 
dition implies that the order of / is a divisor of mn and a mul- 
tiple of m'n, where m' is the quotient obtained by dividing m 
by its largest factor that is prime to n. By means of substi- 
tutions it is easy to show that / may be so determined that its 
order is any arbitrary naultiple of w' that divides mn, when- 
ever the only restriction on / is that the order of /" is n. 

If k is any divisor of g, so that kh=g, there must be at least 
one subgroup of order k in the cyclic group G. This contains 
<t>{k) generators. The theorem that G cannot contain more 
than one subgroup of order k results immediately from the fact 
that the dements of such a subgroup must be as follows: 

since their kih powers are equal to the identity. Hence a cyclic 
group contains one and only one subgroup whose order is any 



50 ABSTRACT GROUPS (Ch. Ill 

given divisor of the- order of the group. In particular, every 
subgroup of a cyclic group is cyclic. 

When g is even, G contains an element of order 2, and every 
element of G which is the square of another element of G is the 
square of exactly two such elements. When g is odd, every 
element of G is the square of only one element of G. In gen- 
eral, the /;th powers of all the elements of G are the elements 
of a group C whose order g' is the quotient obtained by divid- 
ing g by the highest common factor h oi g and k. Each ele- 
ment of G' is the ^th power of exactly h elements of G. The 
continued product of all the elements of G is of order 2 or the 
identity according as g is even or odd. 

If g is the product of two numbers which are relatively prime, 
then G is the product of two cyclic subgroups whose orders 
are these two numbers, and the number of elements of highest 
order in G is the product of the numbers of the elements of 
highest orders in these two cyclic subgroups. This is equivalent 
to the well-known formula that (pirn) = <f){m i) (f)(m2) , whenever 
mi, nii are relatively prime and mim2 = m. The determination 
of all the subgroups of G is equivalent to the determination of 
all the factors of g. If g be written in the form g = P\"p2* 
' Pp"''i Pi, p2, , pp being distinct primes, the number 
of subgroups of G, including the identity, must therefore be 

(ai + l)(a2 + l) . . . (ap+l)-l. 

Since each element of a group generates a cyclic group, 
it is clear that cyclic groups are of fundamental importance. 
When g is a prime number p, every clement of G besides the 
identity generates G, and hence this G docs not involve any 
subgroup besides the identity. The cyclic group of order p 
is therefore the only possible group of this order. That is, 
while there is one and only one cyclic group of every possible 
order, there are orders for which no non-cyclic group exists. 
These orders include all prime numbers. The smallest com- 
posite number which is not the order of any non-cyclic group 
is 15 (cf. Ex. 3, 11). This is a special case of the theorem 
(which will be proved in 70) that two necessary and sufficient 



24] PROPERTIES OF TRANSFORMS 67 

conditions that there is only one group of order g are that g 
is not divisible by the square of a prime number and that none 
of its prime factors is a divisor of the number obtained by 
diminishing by unity another such factor. 

EXERCISES 

1. Prove that the 8 natural numbers which are less than 15 and prime 
to 15 constitute a group with respect to multiplication (mod 15), which 
is not simply isomorphic with the group formed similarly by the numbers 
which are less than 24 and prime to 24. 

2. The smallest group of multiplication which involves the two 
matrices 



1 
0-1 



1 

1 



is of order 8. Is this group simply isomorphic with either of the groups 
of Exercise 1? Find the six other matrices of this group. 

3. To which of the three groups of the preceding exercises is the 
group of movements of the square simply isomorphic? To which is the 
group formed by the 8 numbers less than 20 and prime to 20, with resf>ect 
to multiplication (mod 20), simply isomorphic? 

4. Find two groups whose product is the cyclic group of order 36, 
and determine the number of elements of each order in this group. 

5. Including the identity there are five complete sets of conjugate 
elements in the group of movements of the square. Determine the ele- 
ments of each of these sets. 

6. Do the numbers 2, 4, 6, 8 form a group with resp>ect to multiplica- 
tion (mod 10)? If so, is this group simply isomorphic with the group 
formed by 1, 1, V^Tl, V^? Which of the four numbers 2, 4, 6, 8 
corresponds to the identity? 

7. If 5* is of order 2 find two substitutions of orders 12 and 4 respect- 
ively which may be used for s. 

8. Give the orders of all the pQssible cyclic groups having only two 
distinct generators. 

24. Properties of Transforms. In 9 we considered the 
transform of a substitution. As the concept of transforming is 
very useful we shall develop the properties of this operation 
more fully at this place. Suppose that 

s-Hs = r. 



58 ABSTRACT GROUPS [Ch. Ill 

By raising each member of this equation to the /3th power 
there results the equation 

Hence the theorem: if an element transforms a generator of a 
cyclic group into its ath power it transforms every element of this 
cyclic group into its ath power. 

From the first equation we can also deduce the equation 

s-H^ = tf^. 

Since an element and its transform are of the same order, it is 
necessary that a be prime to the order k of /. Hence If generates 
t, and from Euler's generalization of Fermat's theorem it results 
tKat if m is the lowest power of a such that 

a'" = l(mod k) 

then w is a divisor of 4>{k). Since 5"* is the lowest power of s 
which is commutative with /, it follows that m divides the 
order of s. For instance, an element of order 3 could not 
transform an element of order 5 into any power of itself, except 
the first power, since the numbers 2, 3, 4 belong to the exponents 
4, 4, 2 respectively modulo 5. 

If we form the successive transforms 

5-%5 = /i, S-^tiS = t2, . . . , S-^tn-iS = tn, 

it results that for a sufficiently large value of n we have 

4 = 4, k<n 

since the order of S is finite. This implies that /4-a = /*+, 
a being any positive integer, and if n is the smallest subscript 
for which this relation is true then ^ = 0. That is, the trans- 
forms 

t\, /2, . . . , /o-lj ^ 

repeat themselves in the given order if the powers of s trans- 
form to into n distinct elements. Since 

it results that n must divide the order of s. If this order is a 
prime number />, n is either 1 or p. That is, if s is not com- 



25] CONSTRUCTION OF GROUPS 59 

mutative with t it transforms t into n distinct elements where n 
divides the order of s; whenever this order is a prime number, n 
is equal to the same prime. 

Instead of transforming to by the different powers of s 
we may transform it by all the elements of a group G. If this 
is done, there results a set of elements, each of which has the 
same order, and each is transformed into all the others by the 
elements of G. This set is called a complete set of conjugates 
of /o under G. In particular, all the elements of G can he separated 
into distinct complete sets of conjugates as regards G, and this 
separation can be performed in only one manner. The elements 
of G which transform to into itself form a subgroup of G, and 
the number of the elements in the complete set of conjugates 
to which to belongs is equal to the order of G divided by the 
order of this subgroup. 

By transforming all the elements of a group G by the same 
element / there results a group which is simply isomorphic 
with G, since we obtain a (1, 1) correspondence by making 
each element correspond to its transform with respect to /. 
If t transforms G into itself the resulting simple isomorphism 
is an automorphism of G. 

25. Construction of Groups with Invariant Subgroups. Let 
Si, 52, . . , Sg = G be any group and suppose that / trans- 
forms all the elements of G into elements of G (i.e., t trans- 
forms G into itself), and that T is the lowest power of / which 
occurs in G. The following rectangle 

1, S2, . . ., Sg 



t, S2t, 



p-\ S2P-\ 



., s,t 



/7-1 



is composed of distinct elements since /' cannot be an element 
of the form sj^\ where 8i < 5<7, and all the elements of a row 
are distinct from each other and also from the elements of each 
of the preceding rows. 

In order to prove that the elements of this rectangle repre- 
sent a group, it remains only to show that no additional element 



60 ABSTRACT GROUPS [Cii. Ill 

can be obtained by Combining the elements in every possible 
manner. This fact results from the equation 

Hence the theorem : I J t transforms a group G into itself and if 
/* is the lowest power of t which occurs in G, then t and G generate 
a group whose order is b times the order of G. This theorem is 
very useful in the construction of groups. 

The theorem which has just been proved can be readily 
extended by replacing the cyclic group generated by / by any 
group H. It has been observed that all the common elements 
of G and II constitute a subgroup of both of these groups, 
viz.', the cross-cut of G and H. By replacing the first column 
of the given rectangle by elements from the different co-sets 
of H as regards this cross-cut, we arrive at a more general 
theorem which may be stated as follows: If all the elements 
of a group H transform G into itself, tlien H and G generate a group 
whose order is the order of G multiplied by the index under H 
of the cross-cut of G and H. It is easy to verify that all the 
elements of the group generated by G and H transform G into 
itself. Hence G is an invariant subgroup of this group. 

It was observed in 10 that when 1, S2, Ss, . . . , Sy is 
a subgroup of the group G it is always possible to arrange 
all the elements of G in both of the following ways so that 
no element is repeated: 

1, ^2, S-3, . . ., Sy 1, 52, ^3, . . ., Sy 

t2, 52^2, 53/2, . . -, Syt2 h, t2S2, t^Si, . . ., t2Sy 



h) S2t\, 53/x, . . ., Sytx t\, t\S2, IkSs, > ^X^7 

In these arrangements the elements of the first column do not 
necessarily constitute a group. By interchanging rows and 
columns it becomes evident that such an arrangement is 
possible when the elements of the first row do not form a group. 
The question arises whether all the elements of G can be 
arranged in the given manner even when neither the first row 
nor the first column is a subgroup of G. That such an arrange. 



2Gl THE DIHEDRAL AND THE DICYCLIC GROUPS 61 

ment is sometimes possible follows from the fact that when 
G is the symmetric group of degree 4, we may use for the first 
row the elements of any two Sylow subgroups of order 8, and 
for /2 one of the substitutions of order 4 in the remaining 
Sylow subgroups of order 8. Hence the given rectangular 
arrangement does not always imply that either the first column 
or the first row is composed of the elements of a subgroup. 

26. The Dihedral and the Dicyclic Groups. Let S\ and 52 
represent any two elements of order 2. Their product S\S2 is 
transformed into its inverse by each of the elements S\ and 
52- Each of the elements of the cyclic group generated by 
S\S2 is therefore transformed into its inverse by each of the two 
generators S\ and 52. In particular, this cyclic group is trans- 
formed into itself by each of these generators. Hence two 
elements of order 2 generate a group whose order is twice the order 
of the product of these elements. This group is called the 
dihedral group. When S\S2 is of order 2 the group generated 
by 5i and 52 is the non-cyclic group of order 4. 

There is at least one dihedral group of ever>' even order 
greater than 4, since the group of movements of the regular 
polygon of n sides is clearly such a group. Moreover, there 
is only one abstract dihedral group of order 2, > 1. In fact, 
if there were two such groups their cyclic subgroups of order 
n could be made simply isomorphic. Since each of the remain- 
ing elements of both groups would be of order 2 and would 
transform each element of this cyclic subgroup of order n 
into its inverse, any two of these remaining elements could 
also be made to correspond, and thus a simple isomorphism 
between the two groups could be established. Hence there 
is one and only one dihedral group of every even order greater 
than 2. 

The fact that there is at least one dihedral group of order 
2 can also be easily established by means of substitutions. 
In fact, if n is even we may let 

S\=a\a2-aiaA- . . . -a^.iO^, 

52=0203- . o-20-i- 



62 ABSTRACT GROUPS [Ch. Ill 

If n is odd, these substitutions can be selected as follows: 

52 = 02^3- . . . an-3n-2-7i-l<3fn. 

Since the product of S1S2 in each case is a cyclic substitution 
of order n it results that 5i and 52 generate the dihedral group 
of order 2m. 

The non-cyclic group of order 4 is the only dihedral group 
which does not involve non-commutative elements. A group 
which contains no non-commutative elements is called com- 
mutative or abelian. Since there is one cyclic group of every 
order and one dihedral group of every even order greater than 
2,' there must be at least two groups of every even order greater 
than 2. When this order exceeds 4 one of these two groups, 
whose existence has been proved here, is abelian while the other 
is non-abelian. 

Instead of defining the dihedral group of order 2 as the 
group generated by two elements of order 2 whose product is 
of order , it could also have been defined as the group gene- 
rated by a cyclic group H of order n and an element of order 
2 which transforms every element of H into its inverse. Both 
of these definitions of the dihedral group are very useful. If 
n is even we can find an element / of order 4 which transforms 
every element of // into its inverse and has its square in H. 
The group of order 2w generated by H and this / is called th(^ 
dicyclic group whenever >2, and there is one and only om 
such group of every order which is divisible by 4 and exceeds 
4. The smallest dicyclic group is the group of order 8 generated 
by the four quaternion units 1, i, j, k. This is known as the 
quaternion group. Its properties were studied by W. R. Ham- 
ilton. 

The fact that there is no more than one dicyclic group of a 
given order can be at once proved by proving that two such 
groups of the same order are simply isomorphic. The exist- 
ence of this group for every even value of n may be proved 
by means of substitution groups as follows: Write a transitive 
dihedral group of order 2 on two distinct sets of letters and 



27) REGULAR SUBSTITUTION GROUPS 63 

establish a (1, 1) correspondence between the substitutions. 
Let t be a substitution of order 2 which is commutative with 
each substitution of this dihedral group and transforms the 
two systems of intransitivity into each other. 

The required dicyclic group of order 2 may now be con- 
structed by extending the cyclic subgroup of order n in the 
given dihedral group by means of a substitution of order 4 which 
is the continued product of /, one of two generating substitu- 
tions of order 2 of this dihedral group, and the substitution 
of order 2 which is generated by the cyclic group of order n 
in one of the two transitive constituents of this group. Hence 
there exists one and only one dicyclic group of order 4w, where n 
is any positive integer exceeding unity. 

Every dicyclic group is non-abelian. We have now estab- 
lished the existence of three distinct groups of every order which 
exceeds 4 and is divisible by 4. One of the fundamental prob- 
lems of abstract group theory is a determination of all the 
possible groups of a given order. A considerable number of 
special cases have been solved, but the general solution of this 
problem seems to he far beyond the present developments 
of this subject. As early as 1854 Cayley determined the five 
possible groups of order 8. 

27. Representation of a Group as a Regular Substitution 
Group. Cayley's Theorem. We proceed to prove that every 
abstract group G of finite order can be represented as a regular 
substitution group. Let G=\, S2, . . . , Sg and consider the 
square array of g^ elements formed as follows: 

1, ^2, ^3, . . ., Sf 

52, S2^, 53^2, . . ., SgS2 

53, S2S3, 53^, . . ., SfS3 



Sg, S2St, S3Sg, . . .. Sg 



If we regard the substitutions by means of which each of these 
lines may be obtained from the first, we obtain a substitution 
group on g letters, and each substitution besides the identity 
involves all of these letters. As no two of these substitutions 



64 ABSTRACT GROUPS (Ch. Ill 

are identical, this substitution group is of order g and it is 
simply isomorphic with G. Since each letter of this group U 
replaced once and only once by every other letter, the substi- 
tution group is regular. By combining these facts with those 
of 14 there results the following theorem: Every group of 
finite order can he represented as a regular substitution group, 
and two regular substitution groups which are simply isomorphic 
are also conjugate* 

The fact that every abstract group of finite order can be 
represented as a substitution group enables us to use, in the 
theory of abstract groups, all the theorems of substitution 
groups which are confined to group properties; for instance, 
the theorepis relating to subgroups, quotient groups, sets of 
conjugates, etc. Many of these properties can, however, be 
studied to advantage from the standpoint of abstract groups, 
since we are thus led to fix our attention on the essentials and 
are not distracted by the notation. In some cases, on the 
contrary, this notation appears to be the simplest means to 
establish abstract properties. In fact, we shall see later that 
linear substitution groups also enable us to prove some impor- 
tant abstract group properties very readily. 

From the fact that every group can be represented as a 
regular substitution group it is very easy to derive a simple 
proof of Sylow's theorem. This proof is as follows: 

Let G be any group whose order g is divisible by p" but 
not by /'""^S and represent G as a regular substitution group. 
Suppose that p^ is the highest power of p which is less than g, 
and that g9^p since the case when g is a power of p does not 
require consideration, and consider all the possible substitutions 
on the g letters of G which are of degree p^ and of order p. 
Since G is transitive, it cannot transform any of these substi- 

* This theorem is fundamental, as it reduces the study of abstract groups 
uniquely to that of regular substitution groups. The rectangular array by means 
of which it was proved is often called Cayley's Tabic, and it was used by 
Cayley in his first article on group theory, Philosophical Magazine, vol. 7 
(18.'>4), p. 49. The theorem may be called Cayley's theorem, and it might reason- 
ably be regarded as third in order of importance, being preceded only by the 
theorems of Lagrange and Sylow. 



27) REGULAR SUBSTITUTION GROUPS 65 

tutions into itself. It must therefore transform all of them into 
complete sets of conjugates under G such that each of these 
sets is composed of more than one substitution. As the total 
number of these substitutions is prime to p, according to 14, 
Ex. 7, at least one of these sets of conjugates involves a 
number m of substitutions, where m is prime to p. 

Each of these m substitutions is transformed into itself 
by a subgroup of G whose order is g/m, where m> 1. Hence G 
contains a subgroup whose order is divisible by p". If thb 
subgroup is of order p", our theorem is established. If it is not 
of this order, we have reduced our problem to that of a smaller 
group whose order is divisible by p". In case Sylow's theorem 
were not universally true it would clearly be possible to find 
a smallest group G for which it would not be satisfied. As the 
preceding considerations estabUsh the fact that such a smallest 
group does not exist, they constitute a proof of Sylow's the- 
orem. 

EXERCISES 

1. If a group involves a subgroup whose order is one-half the order 
of the group this subgroup is invariant. 

2. If the order of a group is pq, p and q being prime numbers and p>q, 
this group is cyclic unless p1 is divisible by q. In the latter case there 
are exactly two groups of order pq. 

3. Every simple group of composite order can be represented as a 
non-regular transitive substitution group. 

Suggestion: Consider the substitutions according to which any com- 
plete set of conjugate substitutions or subgroups are transformed under 
the group. 

4. There are exactly two abstract groups of order 4.* 

Suggestion: Represent the possible groups as regular substitution 
groups. 

* The non-cyclic group of order 4 is known under various names. AmonR 
these are the following: Axial grpup, four-group, fours group (Vicrergruppc). 
quadratic group, anharmonic group, and group of the general rectangle. For 
the use of these terms, in order, the reader may consult the following: Picrpont, 
Annals of Mai hematics, vol. 1 (1900), p. 140; Bolza, American Journal of ilatke- 
- -itics, vol. 13 (1891), p. 75; B6cher, Introduction to Higher Algebra, 1907, 
S7; Burnside, Theory of Groups of Finite Order, 1912, p. 444; Capclli, IslUu- 
zioni di analisi algebrica, 1909, p. Ill; Miller, American Maikcmatical itoniUy, 
vol. 10 (1903), p. 217. 



66 ABSTRACT GROUPS [Ch. HI 

6. Every group of order p'^, p being any prime number, is abelian, 
and hence there are exactly two such groups for every prime. 

Suggestion: If such a group is non-cyclic it contains p-\-\ subgroups 
of order p. These could not constitute a complete set of conjugates. 

6. If the order of a group is the double of an odd number, the group 
contains an invariant subgroup of half its own order. 

Suggestion: Write the group as a regular group and observe that it 
contains negative substitutions. 

28. Invariant Subgroups and Quotient Groups. In 13 we 
gave examples of invariant subgroups and of quotient groups 
as related to intransitive substitution groups. The term 
invariant subgroup was defined in 9. Another but equivalent 
definition is based on the following considerations: If H is 
any subgroup of G, then G can be represented in either of the 
following two forms: 

G^H-\-Hs2+ . . . +Hsx 
^H+S2-'H^ . . . ^s^-'H. 

The co-sets HSa, a = 2, . . . , X, are called right co-sets, while 
those of the form Sa~^H are called left co-sets. A necessary 
and sufficient condition that H is an invariant subgroup of G 
is that every right co-set of G as regards H is equal to some 
left co-set of G as regards H. If this condition is satisfied the 
totality of the left co-sets is identical with the totaUty of the 
right co-sets. 

This theorem may be stated more generally as follows: A 
necessary and sufficient condition that H is invariant under the 
substitutions of the right co-set HSa is that Hsa^SaH. When // 
is an invariant subgroup of G, the augmented co-sets of G as 
regards H may therefore be regarded as elements of a new 
group Q, H being the identity of Q* This group Q is the 
quotient group G/H of G as regards H (cf. 13). 

There is an {h, 1) isomorphism between G and Q, h being 
the order of H. When // is the identity this isomorphism 
reduces to a simple isomorphism, and G and Q are the same 

* E. Galois first directed attention to the invariant subgroups and thii: 
important properties. The fact that each invariant subgroup gives rise to a quo- 
tient group is fundamental. 



28] INVARIANT SUBGROUPS 67 

abstract group. Whenever //> 1 the isomorphism is said to 
be multiple. The groups H and Q are called complementary 
groups as regards G, and the product of their orders is equal to 
the order of G. The fact that two different groups may have 
the same complementary groups results directly from the 
dihedral and the dicyclic groups. 

Let q be any element of Q and let s be any one of the elements 
of the corresponding co-set. The order of s must be divisible 
by the order m of q, since s^ is the lowest power of s that occurs 
in H. If m is a power of a prime p then there is an element in 
the corresponding co-set whose order is also a power of p, 
since the group generated by H and this co-set must involve 
a larger subgroup whose order is a power of p than // does. 
Hence the theorem: The order of any element of a quotient 
group divides the orders of all the elements of the corresponding 
co-set, and if this order is a power of a prime number the given 
co-set involves an element whose order is a power of the same prime. 

As a special case of this theorem it may be observed that 
every invariant subgroup of index 2 under any group includes 
all the elements of odd order contained in this group. Two 
elements which belong to the same co-set as regards an invariant 
subgroup are sometimes called equivalent with respect to this 
invariant subgroup. They are also said to be congruent with 
respect to this invariant subgroup as a modulus. It should 
be observed that an invariant subgroup has many of the prop- 
erties of a modulus in elementary number theory. To a smaller 
extent these properties belong to all subgroups, and the terms 
equivalent and congruent are sometimes used in connection 
with any subgroup. 

From the separation of the elements of a group into co-sets 
it results directly that every subgroup of index p under any 
group includes a p'th, p < p, part of the elements of every other 
subgroup of G. We proceed to prove that p<p whenever 
the two distinct subgroups Gi, G2 in question are conjugate 
under G. Suppose that p = p. It must therefore be possible 
to write all the elements of G in the form S1S2, where si 13 any 
element of Gi and S2 is any element of G2. Hence all the con- 



68 ABSTRACT GROUPS (Ch. Ill 

jugatcs of Gi under'G are also conjugates under G2. This is, 
however, impossible, when Hi and ^2 are conjugate since the 
elements of G2 cannot transform d into C2. As the assump- 
tion that p' = p has led to an absurdity, it has been proved 
that the index of the cross-cut of any two distinct conjugate 
subgroups under one of these subgroups is always less than tJie 
index of iJtese subgroups under the entire group. 

To find a simple illustration of this fundamental theorem, 
supp>ose that the index of Gi under G is 2. It follows then 
directly from this theorem that if Gi and G2 are two conju- 
gate subgroups, the cross-cut of Gi and G2 must have an index 
wjiich is less than 2 under each of these subgroups. Hence 
this index is 1, and Gi, G2 are identical. Hence the given 
theorem includes as a special case the theorem that a sub- 
group of index 2 under any group is invariant. This fact can 
also be readily proved in other ways. Cf. preceding Exercises. 

If the invariant subgroup U is composed of all the invariant 
elements of G it is called the central of G and its complementary 
group is known as the central quotient group of G. When this 
central quotient group is abelian G is said to be metabelian* 
The central quotient group is also called the group of con- 
gredient isomorphisms of G. It is clear that the central of G 
is always abelian. For instance, 1,-1 constitute the central 
of the quaternion group, and the central of an abelian group 
coincides with the group. In a non-abelian group the order of 
the central cannot exceed the order of the group divided by 4, 
and if this is the order of the central of G, the central quotient 
group is the axial group. It is easy to prove that tfte central 
quotient group is always non-cyclic. 

29. Commutators, Commutator Subgroup, uid the <^sub- 
group. The element or operator f s'H-^st is called the co7n- 
mutator of s and /, while its inverse is the commutator of / 
and s. When 5 and / are commutative their commutator is 

* W. B. File, Proceedings of the American Association for the Advancement 
of Science vol. 49 (1901), p. 41. 

t The elements of a group are also called operators or operations. We shall 
hereafter use the terms element, operator, and operation interchangeably, since 
all of these terms are commonly found in the modern literature of group theory. 



29] COMMUTATOR SUBGROUP 69 

the identity and vice versa. By writing the commutator 
of 5 and / in the form s-H-^st = c, or t-^st = sc, it is clear that the 
commutator represents an operator which must be multiplied 
into another operator to obtain its conjugate. A group G of 
order g has g^ commutators, but no more than g of them can be 
distinct. In general, these g^ commutators generate a sub- 
group of G, known as the commutator subgroup* of G, or the first 
derived group of G. When G coincides with its commutator 
subgroup it is said to be a perfect group. 

To prove that the commutator subgroup of G is invariant 
under G it is only necessary to prove that the transform of a 
commutator of G as regards any element of G is also a com- 
mutator of G. This fact results immediately from the equa- 
tion 

r-^s-H-^str = r-h-^r-r-H-^r-r-^sr-r-Hr = si-Hi-^Siti. 

It should be emphasized that a group may have many invari- 
ant subgroups, but it has only one commutator subgroup. 
The quotient group which corresponds to the commutator 
subgroup is known as the commutator quotient group. This 
quotient group is always abelian, since the commutator of two 
of its elements must correspond to the commutator of any two 
of the corresponding elements of G and hence it must be the 
identity. 

As the commutator quotient group is abelian, and as every 
invariant subgroup which is complementary to an abelian 
quotient group must include the commutator subgroup, it 
results that this subgroup is the smallest invariant subgroup 
that is complementary to an abelian quotient group. In fact, 
every invariant subgroup which is complementary to an abelian 
quotient group includes the commutator subgroup. From the 
given definition of a perfect group it results also that every 
simple group of composite order is a perfect group, but the con- 

* The term commutator subgroup is due to R. Dedekind, but the fundamental 
properties of these subgroujjs were first published by G. A. Miller, Quarlerly 
Journal of Malhematks, vol. 2S (1S9C.), p. 20<). Their usefulness was quickly 
recognized, and they have entered largely into the recent group theory litcralure. 



70 ABSTRACT GROUPS [Ch. Ill 

verse is not necessarily true. That is, there are perfect groups 
which are not also simple, as we shall see later. 

If the elements of a commutator belong to two invariant 
subgroups of a group G, this commutator is contained in the 
cross-cut of these invariant subgroups. Hence it results 
that if two invariant subgroups of G have only the identity in com- 
mon, every element of each one of these subgroups is commutative 
with every element of the other. 

If the elements of the commutator s'^t-'^st are permuted 
in every possible manner, there result eight operators which may 
be distinct and may all differ from the identity. These eight 
operators are: s-'^t-'^st, t'^sts-^, sts'^t-^, ts-^t-^s, t-^S'Hs, 
st-^s~H, tst-^s-\ s'Hst-^. All of them can be obtained from 
any one of them by means of the substitution group of order 8 
on the four factors. It is evident that each of these 8 commu- 
tators has the same order. 

To prove this it may first be observed that the order of any 
product of n operators is invariant as regards the cyclic group oj 
permutations of these factors, since such permutations are equiv- 
alent to transforming by elements. If reversing the order of 
these n factors does not affect the order of the product, this 
order is invariant as regards the dihedral group of permutation 
of its n factors. In particular, the order of the product of n ele- 
ments of order 2 is always invariant as regards the dihedral group 
of permutations of the n elements. Reversing the order of the 
factors of a commutator cannot affect the order of this commu- 
tator, since it is equivalent to a cyclic permutation of the fac- 
tors of its inverse. 

The given eight commutators, involving s, t and their in- 
verses, are contained in the commutator subgroup of the group 
generated by 5 and /, but they do not necessarily generate this 
subgroup. Since four of them are the inverses of the other 
four, it is clear that no more than four of them are distinct 
when their common order is 2. The most general definition 
of a commutator is, "the product of the transform of an element 
and its inverse." Whenever an element can be written as such 
a product, it may be called a commutator. When we speak 



29] COMMUTATOR SUBGROUP 71 

of the commutator of a group it is assumed that the elements 
of the commutator are elements of the group in question, and 
hence it may happen that only a small number of the elements 
of the group are commutators. For instance, the identity is 
the only commutator in an abelian group. We shall see later 
that every possible group element may be regarded as a com- 
mutator of some two elements. When s and / are both of order 
2 their commutator is the square of their product and is trans- 
formed into its inverse by both of its elements. 

The importance of the concept of commutator is largely 
due to the fact that the commutators of a group represent those 
operators which must be multiplied on the right into a given 
one of a complete set of conjugate operators to obtain all the 
others. Hence the order of the commutator subgroup of a 
group cannot be less than the number of conjugate operators 
in its largest complete set of conjugates. Since the commu- 
tator subgroup is unique, it must evidently correspond to itself 
in every possible automorphism of the group. A subgroup which 
has this property is said to be a characteristic subgroup * Hence 
it results that if a group is not perfect, its commutator subgroup 
is a characteristic subgroup. 

It was observed in 3 that any set of operators belonging 
to any group G of finite order is called a set of independent 
generators of G provided that these operators generate G and 
that none of them is contained in the group generated by the 
rest of them. All the operators of G can be divided into two 
categories, having no common operator, by putting into one 
category all those which occur in at least one of the possible 
sets of independent generators of G, and into the other cate- 
gory those which do not have this property. The operators 
of the second category constitute a characteristic subgroup of 
G, which has been called by G. Frattini the (f>-subgroup of G.f 

If H is any maximal subgroup of G it is evidently always 
possible to select at least one set of independent generators 

* G. Frobenius, Berliner Sitzungsberkhie, 1895, p. 183. 

t G. Frattini, AUi delta Reale Accademia dei Lincei, Rendiconti, scr. 4, vol. 1 

(1885), p. 281. 



72 ABSTRACT GROUPS [Ch. Ill 

of G in such a manner that it includes any arbitrary one of the 
op)erators of G which are not contained in H, while the remain- 
ing operators of the set belong to //. Moreover, there is at 
least one maximal subgroup of G which does not include any 
given one of the independent generators of a particular set of 
independent generators of G. Hence it results that the 0-sub 
group of G is the cross-cut of all the maximal subgroups of G. This 
useful second definition of the (^-subgroup is also due to Frattini. 

If a 0-subgroup of the group G involves a non-invariant 
subgroup or a non-invariant operator, this subgroup or operator 
cannot be transformed into all its conjugates under G by the 
operators of the (^-subgroup. That is, every complete set of 
conjugates of the </)-subgroup is an incomplete set of conju- 
gates under G whenever the former set involves more than one 
element. If this were not the case all the operators of G which 
would transform one of these elements into itself would form 
a subgroup which would not involve all the operators of the 
(^-subgroup of G. This subgroup could not be maximal, since 
it does not involve the 0-subgroup. As any maximal subgroup 
obtained by extending this subgroup by means of operators 
of G could also not involve the (^-subgroup, we have proved 
the theorem: If the <t>-subgroup of a group G involves a non- 
invariant operator or subgroup, the number of conjugates utider 
G oj this operator or subgroup is greater than the number of the 
corresponding conjugates under the <t)-sub group. 

An important special case of this theorem was noted by 
Frattini, who observed that the </)-subgroup of any group involves 
only one Sylow subgroup for every prime which divides the ordci 
of the (^-subgroup. In other words, every 0-subgroup is the 
direct product of its Sylow subgroups, and hence we can alwa} s 
reach the identity by forming successive 0-subgroups, starting 
with any given group. If a group can be represented as a non- 
regular primitive substitution group of degree , its w subgroui)^ 
of degree 1, each being composed of all its substitution- 
which omit a letter, are maximal and have only the identity 
in common. Hence it results that the 4>-subgroup of every 
primitive substitution group is the identity. 



30] SIMPLY ISOMOUPHIC GROUPS 73 

EXERCISES 

1. If an intransitive group of degree h contains exactly it transitive 
constituents, the average number of letters in all its substitutions is nk. 

2. Prove that the group of the square involves an invariant sub- 
group leading to the four-group as a quotient group and that this invariant 
subgroup is its commutator subgroup. 

3. All the elements which are common to all the subgroups of a com- 
plete set of conjugate subgroups constitute an invariant subgroup. 

4. To every invariant subgroup of a quotient group there corresponds 
an invariant subgroup of the group, and to every subgroup which involves 
a given invariant subgroup there corresponds a subgroup in the quotient 
group corresponding to this invariant subgroup. 

5. If every element of a group is raised to the same power and if 
this power is prime to the order of the group, each element of the group is 
found once and only once among these powers. 

6. The commutator subgroup of the symmetric group of degree n is 
the alternating group of this degree, and the alternating group of degree 
n is perfect whenever w>4. 

7. A necessary and sufficient condition that the ^-subgroup of a cyclic 
group is the identity is that the order of this group is not divisible by the 
square of a prime number. 

30. Simply Isomorphic Groups. One of the most miportant 
and most difficult problems in group theory is to determine 
whether twD given groups of the same order are simply isomor- 
phic or not. If they are simply isomorphic they are identical 
as abstract groups and vice versa. Two cyclic groups of the 
same order are always simply isomorphic and a cyclic group 
cannot be simply isomorphic with a non-cyclic group. One 
of the most useful theorems as regards simply isomorphic groups 
may be stated as follows: Two groups of the same order Gi, 
G2 are simply isomorphic if they contain two simply isomorphic 
invariant subgroups Hi, H2 respectively, and are generated by tJiese 
subgroups and two elements /i, /2 such that if tx" is the lowest power 
of t\ which occurs in Hi , then /2 is tlie lowest power of tz that occurs 
in H2, afid /i", to" correspond in the given simple isomorphism 
of Hi, H2. Moreover, it is assumed tliat t\, /2 transform corre- 
sponding generators of Hi, H2 into corresponding elements in the 
given simple isomorphism. 

To illustrate the use of this theorem, let Gi, G2 represent two 



74 ABSTRACT GROUPS [Ch. Ill 

dihedral groups of order 2 and let II i, H2 represent their cyclic 
subgroups of order n. Let tu h represent any two elements 
of Gi, G2 respectively but not contained in Hi, Il2- The common 
order of /i, /2 is 2, and /i, /2 transform corresponding generators 
of III, II2 respectively into corresponding elements, since they 
transform all the elements of these subgroups into their inverses. 
Hence this theorem includes the known theorem that two 
dihedral groups of the same order are always simply isomorphic. 

The given illustrative example of the use of the theorem in 
question may also serve to point out the way toward a proof. 
In fact, if H\, H2 are arranged in a simple isomorphism and 
if the products obtained by multiplying corresponding opera- 
tors by h^, /2^ respectively, /3 = 1, 2, . . . , a\, are placed in 
correspondence, we obtain a simple isomorphism between 
Gi, G2. In fact, /i", t2^ transform all the corresponding opera- 
tors of H\, B.2 into corresponding operators because they trans- 
form generators of B.\, B.2 in this manner. It may be observed 
that this theorem may be employed to prove that two cyclic 
groups of the same composite order are simply isomorphic 
if it is assumed that two cyclic groups of the same prime order 
have this property. 

It results immediately from the given theorem that if two 
abelian groups of the same order involve only operators of the 
same prime order besides the identity, they must be simp!} 
isomorphic. Among the most important simple isomorphisms 
are those in which the operators of the same group G are placed 
into a (1, 1) correspondence in such a way that an automorph- 
ism of G is obtained. We have seen ( 24) that any opera- 
tor which transforms G into itself effects an automorphism on 
its elements. Moreover, any automorphism of G can always 
be brought about by transforming G by some operator which 
transforms G into itself. To prove this statement we shall 
employ a method which has been illustrated in 14 but which 
we desire to exhibit more fully. 

Represent G as a regular substitution group and establish 
an arbitrary automorphism of G. We may suppose that all 
the substitutions begin with the same letter, so that the second 



301 SIMPLY ISOMORPHIC GROUPS 76 

letters of the corresponding substitutions exhibit a substitu- 
tion by means of which we can transform one of these groups 
in such a way that the first two letters of all the corresponding 
substitutions are identical. After this has been done all the 
corresponding letters of the corresponding substitutions must be 
identical, so that the given substitution may be used to effect 
the given automorphism. This follows immediately from the 
fact that the two regular groups in which all of the correspond- 
ing substitutions have the first two letters in common are still 
simply isomorphic, and if in a given substitution c is replaced 
by d, this substitution is the product of the inverse of the sub- 
stitution in which a is replaced by c and the substitution in 
which a is replaced by d. Hence in the corresponding subsU- 
tution of this automorphism c must also be replaced by d, since 
a regular group contains only one substitution in which a given 
letter is replaced by another given letter. 

In view of the importance of this theorem we shall give an 
iUustrative example. Consider the following automorphism 
of the symmetric group of order 6: 



1 


1 


abc'def 


acb-dfe 


ach ' dfe 


abc dcf 


ad-bf-ce 


adbf'ce 


ae-bd-cf 


af-be-cd 


af-be-cd 


ae-bd-cf 



The substitution which transforms the former of these two 
groups so that the first two letters in each pair of corresponding 
substitutions are identical is bc-ef This substitution also 
transforms the former of these corresponding groups into the 
latter. The theorem that every automorphism of G may be 
obtained by transforming G by operators which transform G into 
itself is not only useful in finding the possible automorphisms, 
but it may also be used to determine the totality of the sub- 
stitutions which transform a given group into itself. 



76 ABSTRACT GROUPS [Ch. Ill 

EXERCISES 

1. The substitutions which represent the transformations of the sym- 
metric group of order 6 into all its possible automorphisms constitutes a 
group which is simply isomorphic with this symmetric group. 

2. If a substitution of order 2 transforms d into d it must ako trans- 
form Gi into Gi. 

3. Any group G of order g that involves an invariant operator s of 
order h can be extended by means of an operator / of order nh which is 
commtitative with every operator of G and satisfies the equation t'*=s 
so as to obtain a group of order gn. 

Suggestion: Write G as a regular group on n distinct sets of letters 
and establish a simple isomorphism between these groups. Let /i be 
a substitution of order n which permutes the corresponding letters of this 
intransitive group and is commutative with each of its substitutions. For 
/ we may use the product of h and the substitution s in one of the regular 
constituents. 

31. Group of Inner Isomorphisms. If all the elements of a 
non-abelian group G are transformed by any one of its own 
elements, the elements of G are permuted according to a certain 
substitution. By transforming the elements of G successively 
by all the elements of G there result a series of substitutions 
which constitute a group known as the group of inner isomorph- 
isms oj G. This group is also called the group of cogredient 
isomorphisms of G, and it is simply isomorphic with the central 
quotient group of G ( 28). A necessary and sufficient condi- 
tion that G is simply isomorphic with its group of inner iso- 
morphisms is that the central of G is the identity. 

If G admits isomorphisms which cannot be obtained by 
transforming all the elements of G by its own elements, they 
are called outer, or contragredient, isomorphisms. In this case 
the group of inner isomorphisms is clearly an invariant sub- 
group of the group of isomorphisms of G. 

One of the most useful properties of the group of inner iso- 
morphisms I\ of G is that I\ contains tJie same number of Sylow 
subgroups of every order as G does, provided we call the identity 
a Sylow subgroup of order /> whenever the order of the group 
1 1 is not divisible by p. The truth of this fact becomes evident 
if we observe that a Sylow subgroup of /i has exactly the same 
number of conjugates under /i as the corresponding Sylow 



32) FROBENIUS'S THEOREM- 77 

subgroup of G has under G. In particular, an abeUan group 
contains only one Sylow subgroup of every order. 
- In 13 we defined the term direct product as regards sub- 
stitution groups. In general, a group is said to be the direct 
product of two subgroups which generate it, provided these two 
subgroups have only the identity in common and every element 
of the one is commutative with every element of the other 
For instance, tfie holomorph of a complete group is the direct 
product of tJte group and its coyijoint ( 19). This holomorj^h 
IS also said to be the square of this complete group. A simpler 
lUustration of a direct product is furnished by the axial group 
which IS the direct product of any two of its subgroups of 
order 2. o f 

As the group of inner isomorphisms of G cannot be cycUc 
It cannot have a smaller order than 4. If it is of order 4 G 
contains exactly three abelian subgroups of half its own order- 
and vice versa. When G is non-abeUan and contains more 
than one abelian subgroup of half its order, then G has the 
four-group for its group of inner isomorphisms. As instances 
of groups which have the four-group for their group of inner 
isomorphisms we may cite the octic group and the quaternion 
group. 

32. Frobenius's Theorem. In an article published in the 
Berliner Sitzungsberichte, 1895, page 984, Professor G. Fro- 
benius developed a very fundamental theorem" which may be 
stated as follows: 

// n is any factor of the order g of a group G, the number of the 
operators in G, ifuluding the identity, whose orders divide n, is 
a multiple of n. 

This theorem is evidently true when g is a prime number, 
and also when n=g. Hence, in a proof by complete induction,' 
we may assume that the theorem holds for all the groups whose 
orders involve a fewer number of factors than g does, and also 
for all factors of g which are larger than n. If we can prove 
that it remains true for G and n provided these assumptions 
are made, then the theorem will follow by complete induction 
for an arbitrary group G and any divisor n of its order. 



78 ABSTRACT GROUPS [Cb. Ill 

The theorem is clearly true for every cyclic group, since 
such a group has one and only one subgroup whose order is any 
divisor of the group, and this subgroup includes all the opera- 
tors of this cyclic group whose orders divide this divisor. To 
simplify the general considerations which follow, we shall first 
prove that the theorem applies also to the non-cyclic group of 
order pq, p and q being distinct primes and p>q. Let iV, 
represent the number of operators of G whose orders divide x, 
while Nx represents the number of those operators of G whose 
orders do not divide x. Hence, in the special case when G is 
the non-cyclic group of order pq. 

As Nm is divisible by p we can prove that Np is divisible hyp 
by proving that Np is divisible by p. The fact that Np is divis- 
ible by p results directly from the fact that all of these opera- 
tors are of order q, since G is non-cyclic, and they must occur 
in complete sets of conjugates under a subgroup of order p, 
such that each set involves p of these conjugates. 

We shall now prove, by means of the two given assumptions, 
the general theorem stated above. If p represents any prime 
divisor of g/n, G involves, by h>'pothesis, a multiple of np 
operators whose orders divide np. As 

Nnp = Nn + N'n, 

where N'n represents the number of operators of G whose 
orders divide np but do jiot divide n, it is evident that we 
have only to prove that N'n is divisible by n in order to prove 
our theorem. _ 

The totality of these iV' operators will be represented by 
A. Suppose that n = p^-h where s is prime to />. It is 
clear that A is composed of operators whose orders are divisible 
by p^, and hence it is very easy to see that N'n is divisible by 
p^-^. In fact, every cyclic group whose order is divisible by 
p^ must have a multiple of p^'^ distinct generators, since 
^(^x)=^x-i(^_l)^ ^(^fn) being the totient of m. Hence it 
remains only to prove that N'n is also divisible by s. 



32] FROBENIUS'S THEOREM 79 

Among the operators of A there may be several which have 
the same constituent P of order p^. All such operators are the 
direct product of P and operators whose orders are divisors 
of 5, and all the operators of A may be divided into distinct 
sets such that each set is composed of all the operators of A 
which have the same constituent of order p^. We proceed to 
prove that the number of operators in the combined sets which 
involve all the conjugates of P under G is divisible by 5, and 
hence that the total number of operators in A is divisible by 5. 

To prove this fact, we consider all the operators of G which 
are commutative with P. These form a subgroup H of order 
/>V, and the quotient group of H with respect to the cyclic group 
generated by P is of order r. The orders of the operators of 
this quotient group which divide 5 must also divide the highest 
common factor (/) of j and r. As the order of this quotient 
group involves fewer factors than G does we may assume that 
the number of its operators whose orders divide / is kl. Hence 
A contains exactly kt operators which have the same constit- 
uent P. 

The combined sets which involve no operator of order />^, 
except the conjugates of P under G, must therefore involve 
gkt/{p^r) distinct operators, since P has g/ip^r) conjugates 
under G. Since g is divisible by s and r, it follows that gl is 
divisible by r^ t being the highest common factor of r and s. 
Hence 5 is a divisor of gkt/r. As 5 and p^ are relatively prime, 
5 must also be a divisor of ght/{p\), and the theorem in ques- 
tion has been proved. 

While the number of the operators of G whose orders divide 
any divisor of ^ is always a multiple of n. it does not follow 
that groups exist in which the number of these operators is an 
arbitrary multiple of n. For instance, if p is the highest power 
of the prime p which divides g, G contains at least one subgroup 
of order p", according to Sylow's theorem. If G contains 
only one such subgroup this must be invariant and hence G 
involves only p"* operators whose orders divide p". 

If G contains more than one subgroup of order /)". it must 
contain at least p-\-\ such subgroups, since one such subgroup 



so . ABSTRACT GROUPS (Ch. Ill 

must transform any other into at least p distinct subgroups. 
If G contains exactly /> + l such subgroups they must have in 
common p"'^ operators and hence they involve exactly 

(/>+i)(r-r"')+r"'=r+' 

different operators. Hence G contains exactly p operators 
whose orders divide P" whenever it contains only one subgroup 
of order />", and it contains exactly p"^^ operators whose orders 
divide />* whenever it contains exactly p-\-\ such subgroups. 

We proceed to prove that when G involves more than /> + l 
subgroups of order p", it must also involve more than />"+* 
operators whose orders divide p". 

To prove this fact we may assume that p^ is the largest 
number of operators in common to two subgroups {Hi, H2) 
of order p". Then Hi will transform H2 into p"~^ distinct 
subgroups. At least p-\-l of these have the same subgroup 
of order p^ in common, according to the evident theorem that 
every non-invariant subgroup of a group of order p" is trans- 
formed into itself by at least one of its conjugates. If /3<a 1, 
these p-\-l subgroups involve more than />"+^ operators whose 
orders divide p. If /3=a 1 they must involve exactly p"'^^ 
operators whose orders divide p". If these /> + l subgroups 
are such that they are not transformed among themselves by 
all their operators, there must be another subgroup of order 
P" in G which involves the />"~^ operators common to the given 
p-\-l subgroups, and hence G would involve more than />"+' 
Of)erators whose orders divide p. 

It remains only to consider the case when each of the given 
p-\-l subgroups transforms the remaining p subgroups among 
themselves, and when G contains at least one more subgroup 
of order />". In this case, this additional subgroup of order p" 
would contain operators which would not transform the given 
^+1 subgroups among themselves, and hence G must contain 
more than p" '^^ operators whose orders divide p whenever it con- 
tains more than p-\-l subgroups of order />*. 

This proves that whenever the number of operators in G 
whose orders divide p" exceeds p it must be at least />'*'"*, and 



A 



331 ABSTRACT GROUP REPRESENTED BY SUBSTITUTIONS 81 

hence it establishes the fact that it is impossible to construct 
a group G such that its order g is divisible by p, but not by 
p"'^^ and such that the number of its operators whose orders 
divide p" lies between />* and p"'^^. 

EXERCISES 

1. If the number of the operators of a group G, whose orders divide 
an arbitrary divisor d of the order of G, is exactly d, then G must be 
cyclic. 

2. There are no groups with the property that every cyclic subgroup 
besides the identity is transformed into itself by only its own operators. 
Cf. Dyck, Mathematische Annalen, vol. 22 (1883), p. 101. 

3. Let Xi and n represent two arbitrary numbers, and denote by S\ 
and ^2 the operations of subtracting n from xi and dividing Xi by n respect- 
ively. If n is replaced successively by all the numbers resulting from 
these operations, then Si and ^2 will, tn general, generate the symmetric 
group of order 6 when Xi^=X2. When Xi^=2X'>, and when Xi*=3x2, these 
operations will generate the octic group and the dihedral group of order 
12 resp)ectively.* 

33. Representation of an Abstract Group as a Transitive 
Substitution Group. In 27 it was observed that every group 
of finite order can be represented in one and only one way as a 
regular substitution group. It is often very useful to represent 
a given abstract group as a transitive substitution group on the 
smallest possible number of letters. Many of the abstract prop- 
erties of a group can often be most readily determined if the 
group is written in this form. Hence we proceed to consider 
the general question of representing an abstract group G as a 
non-regular transitive substitution group of degree n. 

In 12 it was observed that when G is thus represented, it 
involves n conjugate subgroups Gi, G2, . . . , Gn each of which 
is composed of all the substitutions of G which omit a given 
letter. If the subgroup of Gi, composed of all its substitutions 
which omit a given letter, is of degree na, these n conjugate 
subgroups are identical in sets such that each set involves a 
of them. As G is non-regular, there must be at least two such 
sets, and hence we see directly that G cannot be represented 

* These groups, together with the four group, have been tailed groups of sub- 
traction and division, Quarterly Journal oj Mathematics, vol. 37 (1906), p. 80l ^ 



82 ABSTRACT GROUPS [Ch. Ill 

as a non-regular transitive substitution group unless it contains 
a non-invariant subgroup H which involves no invariant sub- 
group of G besides the identity. 

This condition is sufficient as well as necessary in order 
that G can be represented as a non-regular transitive substi- 
tution group. In fact, if the elements of G are arranged in a 
rectangle, where those of such a subgroup H appear as the first 
row, as follows: 

5l, S2, . . ., Sn 
Sit2, 52/2, . . ., Snt2 



Sltn, S2tn, , Sf^tn 



the lines are permuted as units if all the elements are multiplied 
on the right by any element of G, since these lines are the co- 
sets of G as regards H. 

Hence each element of G may be denoted by the substitu- 
tion according to which it permutes these co-sets when it is used 
as a multiplier in the given manner. No two elements of G 
could permute these co-sets according to the same substitution, 
since H is non-invariant under G and does not involve any 
invariant subgroup, besides the identity, of G. This proves 
the following theorem: A necessary and sufficient condition 
that an abstract group G of order g can be represented as a transitive 
substitution group of degree n is that G contains a non-invariant 
subgroup of order g/n which does not include any invariant sub- 
group of G besides the identity. 

In the given method of representing G as a transitive sub- 
stitution group of degree n it is clear that H corresponds to the 
subgroup composed of all the substitutions which omit a given 
letter of this transitive group. All the subgroups of G which 
correspond to // in one of the possible automorphisms of G 
give rise to the same transitive substitution group, but no other 
subgroup can have this property. Hence G gives rise to as 
many different transitive groups of degree n as it has sets of 
subgroups of order g/n such that each set includes all those 
subgroups which correspond in some one of the possible auto- 



33] ABSTRACT GROUP REPRESENTED BY SUBSTITUTIONS 83 

morphisms of G, and such that these subgroups do not include 
any invariant subgroup of G besides the identity.* 

For instance, the symmetric group of order 24 contains three 
conjugate cyclic subgroups of order 4 and also three conjugate 
noncyclic subgroups of this order. It contains no other non- 
invariant subgroup of order 4. Hence this symmetric group 
appears twice among the transitive substitution groups of 
degree 6. As the alternating group of degree 4 contains two 
sets of conjugate subgroups, of orders 2 and 3 respectively, 
this group can be represented transitively in only two ways 
besides the regular form. That is, it appears as a transitive 
group of degree 4 and also as a transitive group of degree 6. 
It should be observed that these considerations establish another 
ver>- close contact between the theory of abstract groups and 
that of transitive substitution groups. 

It is now easy to prove the theorem, to which we referred 
in 10, that the co-set multipliers may be so selected that they 
are the same on the right as on the left. When the subgroup 
// is invariant this requires no proof. When H does not involve 
any invariant subgroup of G besides the identity, and G is 
represented as a transitive substitution group of degree n with 
respect to H, it may be supposed that H is composed of all the 
substitutions of G which omit a given letter a. The right 
co-sets will then be composed separately of all the substitutions 
of G in which a is replaced by a given letter. In the left co-sets 
a is replaced by all the letters of a transitive constituent of H 
if H is intransitive on 1 letters. If H is transitive on n 1 
letters the theorem is ev^dent^ 

Hence it remains only to consider the case when H is intran- 
sitive. In this case it is evidently possible to select a certain 
number of left co-sets involving all the substitutions of the 
same number of right co-sets, and in these the multipliers 
may be made the same in the left co-sets as in the corresponding 
right co-sets. Hence the theorem is established in case H con- 
tains no invariant subgroup of G besides the identity. 

If // involves such an invariant subgroup but is not itself 
Bulletin of the American Mathematical Society, vol. 3 (1897), p. 215. 



84 ABSTRACT GROUPS [Ch. Ill 

invariant under G, H will correspond to a subgroup in a quotient 
group, such that this quotient group can be represented trans- 
itively with respect to this subgroup. Hence the theorem 
is also true in this case. For an abstract proof of this theorem 
the reader may consult H. W. Chapman, Messenger of Mathe- 
matics, vol. 42 (1913), page 132. 

EXERCISES 

1. If a dicyclic group is represented as a transitive substitution group 
it must be regular. 

2. A dihedral group of order 2m, n>2, can be represented in two and 
in only two ways as a transitive substitution group. 

3. Only one of the five possible groups of order 8 can be represented 
as a non-regular transitive substitution group. 

4. If a simple group of composite order is represented as a transitive 
group of lowest possible degree it must be primitive. 

5. There are five and only five abstract groups of order 12. 

34. Historical Note.* The concept of group is one of the 
oldest mathematical concepts. Even in the development of 
elementary geometry by the Greeks this concept played a 
fundamental role, as was pointed out by H. Poincare in an 
article entitled " On the foundations of Geometry," Monisl, 
volume 9 (1898), pages 1-43. It was, however, not developed 
into an extensive theory until a comparatively recent period. 

In the latter half of the eighteenth century various writers, 
especially J. L. Lagrange and A. T. Vandermonde, began to 
lay stress, in their algebraic investigations, upon the elements 
of the theory of substitutions. On the other hand, L. Euler 
brought some properties of abelian groups into prominence, 
especially by his work on power residues. Towards the close 
of the eighteenth and at the beginning of the nineteenth cen- 
tury, two Italian mathematicians, P. Rufl5ni and P. Abbati, 
entered more directly on the study of substitution groups 
by proving that there are no three or four-valued rational 
functions of more than four variables, and that a two-valued 
function must be alternating. 

During the first half of the nineteenth century a number 
of other investigators entered this field. Foremost among 

Cf. Bibliolheca Malhemalica, vol. 10 (1910), p. 317. 



34] HISTORICAL NOTE ON GROUPS 86 

these were C. F. Gauss, A. L. Cauchy, N. H. Abel, and E. 
Galois. The first of these helped to lay a foundation for abelian 
groups by his investigations in number theory, while the other 
three contributed directly towards the development of the 
general theory of substitution groups. Abel and Galois solved 
two fundamental problems in the theory of equations by means 
of substitution groups, and thus they directed attention to the 
usefulness of this subject (Cf. Part III). On the other hand, 
Cauchy ordered and extended the results obtained by his 
predecessors and contemporaries, and laid a broad foundation 
for the general theory of substitution groups. Hence Cauchy 
is frequently called the founder of this theory. 

The theory of abstract groups grew gradually out of that 
of substitution groups. In fact, we find even in Cauchy's later 
writings a tendency to state some results independently of the 
notation. Cayley's table and Cayley's theorem are very 
fundamental in this theory, and hence A. Cayley is sometimes 
called the founder of the abstract theory of groups. In 1856 
W. R. Hamilton gave abstract definitions of the groups which 
are simply isomorphic with the groups of movements of the 
five ancient regular solids. By these definitions, and the fact 
that his quaternion units and their negatives form an impor- 
tant non-abelian group of order 8, he contributed considerably 
towards an interest in this theory. 

A solid foundation for an abstract theory implies, however, 
clear abstract definitions of the terms used. Among the earlier 
writers one fails to find such definitions. According to E. V. 
Huntington, Transactions of the American Mathematical Society, 
volume 6 (1905), page 181, the earliest explicit set of postulates 
for abstract groups were given by L. Kronccker in 1870. Even 
at the present time the term group is sometimes used with 
different meanings as a mathematical technical term.* 

The earliest separate text-book devoted to group theory 

is Jordan's Traite des substitutions et des equations algibri- 

ques, Paris, 1870. This work has become a classic. It was 

written, however, before the abstract theory was well estab- 

* Cf. Encycloptdie des Sciences Malhimatiques, tome 1, voL 1, p. 576; vol. 2, p. 243. 



86 ABSTRACT GROUPS (Ch. Ill 

lished. Even Sylow's theorem appeared two years later, and 
a number of other more recent theorems make it possible to 
present many subjects in a simpler manner than could be done 
at the time when Jordan wrote this Traite. The only other 
separate text-books on groups of finite orders which appeared 
before the present century are Nctto's SubstitutionentJieorie, 
1882, Burnside's Theory of Groups of Finite Order, 1897, and 
Bianchi's Lezioni sulla theoria dei gruppi di sostituzioni, 1897 
(lithographed). Netto's book was translated into Italian by 
G. Battagb'ni in 1885, and into English by F. N. Cole in 1892. 

In the older writings on groups, and in some of the more 
recent ones, a set of distinct elements considered with respect 
to only one formal law of combination is said to form a group 
whenever every element obtained by combining an element with 
itself or with any other element of the set is again in the set. 
In most cases it is, however, tacitly assumed that these elements 
obey the other laws which were imposed in our definition of a 
group. If this is not assumed the set may be said to have 
" the group property." Cf. M. Bocher, Introduction to Higher 
Algebra, 1907, page 82. 

Since the beginning of the present century a considerable 
number of separate treatises on groups of finite order have 
appeared. The literature on this subject has been made more 
accessible by the publication of two somewhat extensive bib- 
liographies, viz., The Constructive Development of Group-Theory 
by B. S. Easton, University of Pennsylvania, 1902, and the 
" Essai d'une bibliographic sur la theorie dcs groupes " by C. 
Alasia, published in the Rivista di fisica, matltematica e scicnze 
naturali, Pavia, 1908-10. 

The main theorems of the theory of finite groups, together 
with numerous historical data relating to their development 
may be found in the Encyclopedie des Sciences Mathematiques, 
tome 1, volume 1, page 532, and in the second edition of Pas- 
cal's Repertorium der hdheren Mathematik, volume 1, page 168. 
A list of remarks on the bearing of the theory of groups, 
exhibiting the wide application of this subject, was published 
in volume 6 (1914), of the Tohoku Mathematical Journal. 



CHAPTER IV 
ABELIAN GROUPS * 

35. Invariants. A group is said to be abelian when each of 
its elements is commutative with every element of the group, 
26. Since all the Sylow subgroups of the same order are 
conjugate under every group, it results that an abelian group can 
have only one Sylow subgroup of a given order, and hence e.zry 
abelian group is the direct product of its Sylow subgroups when- 
ever its order is divisible by more than one prime number. This 
implies that a necessary and sufficient condition that two abel- 
ian groups are simply isomorphic is that all their Sylow sub- 
groups are simply isomorphic, and hence the study of abelian 
groups is reduced to the study of such groups whose orders are 
powers of a single prime number. In particular, if the order of 
an abelian group is not divisible by the square of a prime number 
the group must be cyclic. 

Suppose that G is an abeUan group of order />", p being any 
prime number. If G is cyclic, all of its elements are generated 
by a single one of them si, where Sy may be selected in ^'" />""* 
ways. Moreover, every element of order p", m>a>0, in G 
is the pth power of p distinct elements of order p"'^^, the p^th 
power of p^ distinct elements of order p"'^'^, the p^'"~''hh power 
of p^'" distinct elements of order />*". When G is not cyclic 
we may choose for si any one of its elements of highest order. 
Some power of every other element of G is contained in the group 
(si) generated by si. We represent by S2' one of those elements 
which have to be raised to the highest power p"' to obtain an 

* A number of the theorems on abelian or commutative groups were first 
developed by L. Eulcr and C. F. Gauss in connection with their studies in number 
theory. The earliest extensive exposition of the properties of these groups is 
due to G. Frobenius and L. Stickclbcrger, Crelle, vol. 86, p. 217. 

87 



88 ABELIAN GROUPS (Ch. IV 

clement of (^i). That is, the p"th power of every element 
of G is in (51) but the />"'"' power of S2' is not in (si). We 
proceed to prove that S2' may be so selected that the cyclic 
groups (51) and {s'2) have only the identity in common; that is, 

Since 52''^ is in (^i), and the order of 52' does not exceed 
that of ^1, there must be an element in (si) whose p^^th power 
is the inverse of 52'*""'. The product of S2' and this element is 
therefore of order p'\ Moreover, the /?'~Hh power of this 
product is not contained in (si) since one of its factors is in 
(si), while the other does not have this property. In what 
follows we shall denote this product, of order p*, by 52. If 
the order of G is the product of the orders of Si and S2 it is clear 
that G is the direct product of (51) and (52). We proceed to 
prove that G is always the direct product of cyclic groups.* 
The orders of these cycUc groups are called the invariants f 
of G. In particular, if si is of order />" and if G is the direct 
product of (51), (52), the invariants of G are />"', />". 

If G contains elements which are not included in the group 
(51, S2), generated by si and ^2, we may suppose that the p*th 
power of every other element is in (51, S2) while the />"*"* th 
powers of some of the elements are not in this subgroup. We 
proceed to prove that at least one of the latter elements is 
of order P"*. Let S3' be any one of these elements. As 

is in (51, S2), and as a3<a2<ai, there must be some ele- 
ment in (^1, S2) whose />"'th power is the inverse of s^'""'. 
The product, 53, of this element and j'3 is therefore of order p"', 
and G involves the direct product of the three groups (^i), 
(^2), (sz)- As this process may clearly be continued until all 

* It is implied that each of these cyclic groups has only the identity in common 
with the group generated by all the others. In other words, they are independent 
cyclic groups. 

t The invariants of an abclian group have also been called the elementary 
divisors of the order of this group. Frobenius and Stickelberger, Crelle, vol. 
86 (1878), p. 238. 



35) INVARIANTS OF ABELIAN GROUPS 89 

the elements of G have been exhausted, it has been proved that 
every noti-cyclic abelian group of order p^ is tlie direct product of 
independent cyclic groups. This is the most important theorem 
relating to abelian groups.* 

These independent cyclic groups may be represented as 
substitution groups on distinct sets of letters. Moreover, 
it is clear that a group can be constructed such that the orders 
of such substitution groups are arbitrary, and hence the prod- 
uct is an abelian group of arbitrary order. That is, the number 
of distinct abelian groups of order />" is equal to the number of the 
possible partitions of m as regards addition, and each of these 
groups may be completely defined by the value of p and the 
s>Tnbol (mi, W2, , Wx) where mi, m2, . . . , m\ repre- 
sent positive integers such that W1+W2+ . . . -\-m\ = m. The 
group G is completely defined by p and the values of the inte- 
gers mi, mo, , ffi\, and it is not affected by the order in 
which these integers are arranged. It is said to be of order p^ 
and of type (mi, W2, . , m-y). We may therefore suppose that 
the numbers mi, m2, - . . , mx are always arranged in order 
of magnitude, beginning with the largest. It may be added 
that the determination of the number of possible abelian groups 
of order p^ is reduced by these theorems to a problem in the 
theory of numbers; viz., the determination of the total num- 
ber of possible partitions of m as regards addition. This prob- 
lem has received considerable attention, but it still involves 
many unsolved difficulties. 

Suppose that an abelian group G of order /> has or invariants. 
All of its elements whose orders divide p must constitute a sub- 
group of order p"; and, conversely, whenever these elements 
constitute a group of order p", G has exactly a invariants. The 

* A set of independent generating elements can generally be selected in a 
large number of ways. Such a set is often called a base of the abelian group, 
and the operators si, Si, 5j, . . . are called elements of the base. The funda- 
mental theorem that every abelian group is a dircA product of independent 
cyclic groups is implicitly contained in the works of C. F. Gauss and K. Schering, 
but L. Kronecker gave the first satisfactory proof of it in 1870. It should periiaps 
be placed next after Cayley's theorem among the most fundamental theorems 
of group theory. 



90 ABELIAN GROUPS [Ch. IV 

elements of G whose orders divide p^ constitute a group of 
order ^^**~^, where /3 is the number of invariants of G which 
are equal to p. In general, let Xi, X2, . . . , x^ represent the 
number of the invariants of G which are equal to p, p^, . . . , p^ 
respectively, and supp)Ose that they include all the invariants 
of G. The number N of the elements of G whose orders divide 
/>*, /fe < X, is then given by the formula 

It should be observed that the orders of the independent 
cyclic groups which generate a given group G are completely 
determined by G when G is of order />"*. In general, G is the 
direct product of Sylow subgroups and hence it is also the 
direct product of a series of cyclic independent subgroups 
Ci, C2, . . . , Cm\ each of which has for its order a power of a 
prime number. Unless the contrary is stated it will be assumed 
that the order of each of these subgroups exceeds unity, and 
hence their number and their orders are completely determined 
by G; and, in turn, they determine G completely. That is, 
if these orders are the same for two groups these groups are 
simply isomorphic. These orders are therefore invariants 
of G, but they are not the only numbers which are known as 
invariants of G. Their number constitutes the largest possible 
number of orders of independent cyclic groups in G; that is, 
neither G nor any of its subgroups can have more than m' 
independent generators. 

It is important to note that the term set of independent gen- 
erators as regards abelian groups is usually employed to repre- 
sent a set of independent generators which is such that the 
group generated by an arbitrary number of them has only the 
identity in common with the group generated by the remaining 
ones. In dealing with abelian groups we shall always use this 
term with this special meaning and not with its general meaning 
given in 3. 

36. Largest and Smallest Number of Possible Invariants. 
We proceed to find the smallest number of independent gen- 
erators of G. The given subgroups Ci, C2, , Cm' can be 



36] NUMBER OF POSSIBLE INVARIANTS 91 

arranged in rows such that the orders of all those in one row 
are powers of the same prime, and such that the order of each 
is equal to or greater than the order of the one which follows 
it in the same row. In case the rows do not contain the same 
number, the vacant places may be filled by the identity. 
Arranging these rows in the form of a rectangle, we have 

Cl, C2, . . . , Ca 

^o+l> ^o+2> ^2a 



By forming the products of all those in each column we 
obtain a independent cyclic subgroups such that the order of 
each is divisible by the orders of all those which follow it. 
These subgroups form the smallest possible number of generating 
cyclic subgroups of G. The orders of these subgroups are com- 
monly called the invariants of G, since any other system of 
independent generating subgroups in which the order of every 
group is divisible by the order of every following group is 
composed of groups which are simply isomorphic with these 
products. It may be observed that a is the largest number of 
invariants in a Sylow subgroup of G, while m' is equal to the 
sum of the numbers of invariants of all the Sylow subgroups of 
G. It is clear that the independent generators of G can be so 
selected that their number has any arbitrary value from a to 
m' , but this number can have no other value. Moreover, G 
cannot be generated by less than a cyclic subgroups even if 
these subgroups are not independent. 

Whenever the independent generators of G are so chosen 
that the order of each of them is divisible by the orders of all 
those which follow it, their number must be a, and when the 
order of each is a power of a single prime their number must 
be m\ but it is not true that the independent generators can be 
so arranged that the order of each is divisible by the order of all 
those which follow it whenever the number of these generators 
is a. The two numbers a and m' are only equal when the order 
of G is a power of a single prime. Since the former method 
leads to the smallest number of invariants it seems appro- 



I 



92 ABELIAN GROUPS [Ch. IV 

priate to call the orders of these independent generators the 
invariants of G, although the latter method has some advan- 
tages. The choice of invariants such that their number lies 
between a and m' seems less natural. Wc evidently arrive 
at the a invariants if we choose the independent generators 
in the following way. Start with an element of highest order 
and then select any other element such that the two generate 
the largest possible subgroup. The orders of two independent 
generators of this subgroup are the first two invariants of G. 
If we add to this subgroup another element so that the three 
generate the largest possible subgroup, we arrive at the third 
invariant, etc. 

A marked difference between the two given methods of 
arriving at the invariants of G should perhaps be emphasized; 
viz., the orders of the independent generators of G are completely 
determined by m\ but not by a. That is, if two sets of m' 
independent generating cyclic subgroups of G were given, the 
orders of the subgroups of one set would be the same as those 
of the other; but if two sets of a independent generating cyclic 
subgroups of G were given, the orders of those of one set could 
generally vary a great deal from the orders of those of the 
other. A necessary and sufficient condition that the orders of 
these two sets must be the same is that the a invariants of each 
of the Sylow subgroups of G, with one possible exception, are 
equal. The number a is said to be the rank of G. 

If G is the direct product of a series of subgroups Gi, G2, 
. . . , G\, we may select a set of independent generators of G 
by combining arbitrary sets of independent generators of each 
of these subgroups. Suppose that Gi, G2, . . . , G\ are the 
Sylow subgroups of G. Any element t oi G will have a constit- 
uent, which may be the identity, from each one of these sub- 
groups, and the order of / will be the product of the orders of 
these constituents. To determine the number of the elements 
of a given order in G it is only necessary to determine the num- 
ber of elements of a given order in each of the Sylow subgroups. 
That is, if the order of / is />i'"/>2'^ px"^ (p\, p2. . . . , px 
being prime numbers), the number of elements of G whose 



37l NUMBER OF ELEMENTS OF A GIVEN ORDER 93 

order is equal to the order of t is the product of the numbers 
of the elements of orders />i*, p2*, , px'^^ in the respective 
Sylow subgroups of G. We proceed to determine this number. 
37. Number of Elements of a Given Order. Let G be any 
abelian group of order />*" whose invariants are />"', p*^, . . . , 
^"^(ai f a2 > as > . . . >ax>0). Let- m'i = \ represent the 
number of invariants ^ p, m'2 the number of those ^ />2, . . . , 
and m'a^ the number of those =/>"*. To determine the num- 
ber of the elements of order />^(l^/3^ai), it is only necessary 
to find the order of the group generated by all the elements 
whose orders divide />" and to subtract from this number the 
order of the group formed by all the elements whose orders 
divide p^~^. That is, the number of elements of order p^ 
in G is equal to * 

To obtain the number of the elements of a given order n 
in any abelian group we may wTite n in the form 2'pi*p2'^ 
. . . Px"'^, and find the number of the elements of order 2** 
in the Sylow subgroup of even order, then find the number of 
the elements of order />i"' in the Sylow subgroup whose order is 
divisible by pi, etc. The product of all the numbers obtained 
in this way is equal to n. For instance, to find the number of 
the elements of order 12 in the abelian group whose invariants 
are 24, 6, 2, we observe that the invariants of the Sylow sub- 
groups are 8, 2, 2 and 3, 3 respectively. The number of 
the elements of order 4 in the former Sylow subgroup is 8, 
(m'l = 3, m'2 = 1, ^'3 = 1), and the number of those of order 3 in 
the latter is also 8. Hence there are exactly 64 elements of 
order 12 in the given abelian group. 

The number of the elements of order p^ in the group G of 
order />"* may also be obtained by observing that if 5i, 52, . . . , 
5x represent a set of base elements of G, a set of base elements 
of the subgroup of G generated by all its elements whose 

HefTter, Crellc, vol. 119 (1898), p. 261; Netto, VorUsungen ilbcr Algebra, 
vol. 2 (1900), p. 248. 



h 



94 ABELIAN GROUPS [Ch 1 

orders divide ffi may be obtained by raising all those of the 
given set whose orders exceed p^ to a power sufficient to reduce 
their orders to p^. The elements whose orders divide p^'^ 
constitute a subgroup of index p'^'fi under the given subgroup 
of G. Hence the number of elements of order ^ is p^'fi\ 
times the order of the subgroup of G which is composed of all 
its elements whose orders divide /?^~^ 

EXERCISES 

1. Determine the number of elements of each order in the four abelian 
groups of order 100. 

2. The smallest number of letters on which an abelian group can be 
represented as a substitution group is the sum of its invariants, if all these 
invariants are powers of prime numbers. 

3. A group of order p''q must be abelian when ^ is a prime number 
which is less than the prime number p and does not divide p-l. 

4. If all the elements besides the identity of a group are of order 2 
the group is abelian. 

5. The order of the <^-subgroup of any abelian group of order g is equal 
to g divided by the order of the subgroup generated by all the operators 
of prime orders contained in this abelian group. 

38. Abelian Groups of Given Orders. The number of the 
different possible abelian groups of order n = 2'^pi'^p2^ . . . p^"^ 
(pi, p2, , p\ being distinct odd primes) is equal to the 
product obtained by multiplying together the numbers which 
separately represent the total numbers of partitions, as to addi 
tion, of the separate exponents ao, ai, "2, ,ax which exceed 
zero. In particular, if none of these exponents exceeds 3 the 
number of distinct abelian groups of order n is equal to the 
product of those exponents which exceed zero. For instance, 
the number of abelian groups of order 32- 5^- 7-^ is 18, while 
the number of those of order 2'^ -32 -5^ is 7 2-5 = 70. 

From the preceding paragraph it results that it is very 
easy to determine the number of the possible abelian groups 
of a given order. It may be observed that two abelian groups 
which involve the same number of elements of each order are 
simply isomorphic, but this is not true as regards non-abeUan 
groups. In fact, we can easily construct non-abelian groups 
which have the same number of elements of each order as 



i 



39] SPECIAL CI^SS OF ABELIAN GROUPS 96 

certain abelian groups. For instance, if 5 is an element of order 
8 while / is of order 2, the two groups of order 16 generated by 
s and t, when these elements satisfy one of the following 
equations 

tst=s, tst=s^, 

evidently contain the same number of elements of each order. 

To prove that two abelian groups which have the same 
number of elements of each order are always simply isomorphic, 
it is clearly only necessary to consider the case when their 
order is a power of a prime. In this case, it is easy to see that 
any change in the invariants will affect the number of elements 
of given orders. In fact, if the order of such a group is p"*, 
it has been observed that the elements of order p generate a 
group of order p, where a is the number of its invariants. 
The elements of order p^, if there are such, generate a group of 
order p^~^ where ^ is equal to the number of invariants which 
are equal to p, etc. Hence it results that two abelian groups 
which have the same number of elements of each order are simply 
isomorphic. 

39. A Special Class of Abelian Groups. We gave, in 5, 
illustrations of abelian groups which are generated by the 
0(g) totitives of g, that is, by the <f>{g) natural numbers which 
do not exceed g and are prime to g, and we shall now enter upon 
a more detailed study of these important abelian groups.* 
It is easv to s ee that they c onstitute the groups of isomorphisms 
of cyc lic groups ; that is, the groups according to which the 
elements of cyclic groups are permuted when these cyclic groups 
are made simply isomorphic with themselves in every possible 
manner. For the sake of simplicity we begin with two illus- 
trative examples. Let 1, a, o?, o-"^, a^, represent the five fifth 
roots of unity. These may be put into a (1, 1) correspondence, 
or they can be made simply isomorphic with each other, in the 
following four ways, but in no other way: 

* In the second edition of vol. 2 of Weber's Lekrbuch der Algebra, 1809, 
j.. (,(), these groups are called the most important example of abelian groups of 
finite order. 



96 ABELIAN GROUPS (Ch. IV 



1 1 


1 1 


1 1 


1 1 


aa 


aa2 


ac? 


a a"* 


o^o^ 


a2a4 


o?a 


o?o? 


o?o? 


o?a 


o^a^ 


c?o? 


o^c^ 


a'^o?. 


o^o? 


o^a 



As the different ways in which a group can be made simply 
isomorphic with itself clearly correspond to the substitutions 
of a group, it results that the group of isomorphisms of the 
group of order 5 is of order 4. As a substitution group whose 
elements are the numbers 1, 2, 3, 4 the given group of isomorph- 
isms may be represented in the following manner: 

1, 1243, 1342, 14. 23. 

That is, the group of isomorphisms of the group of order 5 is 
the cyclic group of order 4. The same abstract group of order 
4 is evidently generated by the following four numbers, when 
they are combined by multiplication: ^ 

12 3 4. (mod 5); 

As a second illustrative example we consider all the possible 
simple isomorphisms between the eight eighth roots of unity 
represented by the following symbols: 1, /3, /S^, /3^, ^, j8^, /3^, /3", 
where ^ is any root of the equation ic* + 1 = 0. 
isms may be represented as follows: 



1 1 


1 1 


|8/3 


/3/33 


/32/32 


/32/36 


/33^ 


/33/3 


^^ 


^/3* 


fi^0^ 


/3S^7 


jge/S* 


^6^2 


^7^7 


^7^5 



::4 + l=0. 


These isomorph- 


1 1 


1 1 


^/3^ 


/3^7 


^2^2 


/32^6 


/3307 


/33/35 


^^ 


^^ 


/3/3 


^5^ 


^6^6 


^6^2 


/37^ 


^70 



This group of isomorphisms, as a substitution group on the 
numbers 1, 2, . . . , 7, is as follows: 

1 13-57-26 15-37 17-35-26. 



y 



39] SPECIAL CLASS OF ABELIAN GROUPS 97 

This group could have been equally well represented by 
1 13-57 15-37 17-35 

and it is clear that it is simply isomorphic with the group formed 
by the following numbers, when they are combined by multi- 
plication : 

13 5 7 (mods). 

These examples may sufl&ce to illustrate the fact that the 
group formed by the <^(w) totitives, with respect to multiplica- 
tion (mod w), is the group of isomorphisms of the cychc group 
of order m. To prove this fact it is only necessary to observe 
that the correspondence of the operators of lower orders in a 
cyclic group is completely determined by the correspondence 
of the operators of highest order, and that all of the latter may 
be obtained from any one of them by raising it to all the various 
powers which are prime to the order of the cyclic group. In 
particular, a necessary and sufficient condition that the num- 
ber m has a primitive root is that the group formed by the 
<^(w) totitives (mod w) be cyclic. While the group formed by 
the totitives is always abelian, there are many abelian groups 
which cannot be represented in this way. Hence these groups 
form a special class of abeUan groups. 

We proceed to determine some conditions which must be 

satisfied in order that an abelian group G may belong to this 

class. When G is cyclic the matter is quite simple. It is 

necessary and sufficient that its order g be the exponent to 

which a primitive root of some number belongs. That is, 

whenever g = p^{pl),* ^ being an odd prime number, and a 

being any positive integer or zero, the cyclic group G belongs 

) the given class, and only then. The lowest two even num- 

>v.'rs which are not of the form />"(/> 1) are 8 and 14; hence 

these numbers are the lowest orders of cyclic groups of an 

even order that cannot be the groups of isomorphisms of any 

cyclic groups whatever, and hence the cyclic groups of these 

two orders cannot be represented as groups of totitives. 

* Cf. Dirichlet-Dedekind, Zahkntheorie, 1879, p. 340. 



: ABELIAN GROUPS [Ch. IV 

If g is written in the form g = 2"/>i'"/>2' . {pi, p2, . - 
being different odd prime numbers); G is the direct product 
of its subgroups of orders 2**, pi', p2*, . . , and its group 
of isomorphisms / is evidently the direct product of the groups 
of isomorphisms of these subgroups. Since the group of iso- 
morphisms of a cyclic group whose order is a power of an odd 
prime number is cyclic, it follows from the above that / is the 
direct product of the cyclic groups of orders />]"" *(/>i l), 
/>2***~*(/>2 1), , when 00 = or 1. When ao>l, we have 
to add a group of order 2 and a cyclic group of order 2'^'^ to 
these factor groups in order to obtain /, since there are numbers 
which belong to the exponent 2*^"^ (mod 2"), but none which 
belong to a higher exponent.* 

Since / is the direct product of groups of even orders, the 
order of / is always even when g>2. It can clearly be any 
even number of the form 2^pi^'p2^ . . . {pi-l){p2-l) . . . 
The smallest two natural numbers which are not of this form 
are 14 and 26 ;t hence these numbers are the lowest orders of 
groups that cannot be groups of isomorphisms of any cyclic 
group whatever. It is evident that the highest prime factor 
of the order of / can not exceed the highest prime factor of g. 

EXERCISES 

1. Determine the invariants of the group formed by the 40= (^(100) 
totitives of 100 (mod 100). 

2. The number of invariants in the group of the totitives of m (mod 
m) is equal to the number of the distinct odd prime factors of m when- 
ever m is either odd or double an odd number. It is equal to the num- 
ber of distinct prime factors of m, whenever m is divisible by 4 but not 
by 8; when m is divisible by 8, the number of these invariants is one 
more than the number of the distinct prime factors of m. Cf. Weber, 
Lehrbuch der Algebra, vol. 2, 1896, p. 59. 

3. Find the three possible cyclic groups whose group of isomorphisms 
has the invariants 6, 2, 2. 

4. If the operators of order 2 in the group of isomorphisms of the cyclic 
group of order m generate a group of order 2", what is the maximum num- 
ber of the distinct primes which divide w? What is the minimum number 
of such divisors of m? 

* Cf. H. VVcbcr, Lehrbuch dcr Algebra, 2d cd., vol. 2, 1899, o. 64. 
t Lucas, Thiorie dcs nombrcs, 1891, p. 394. 



40] SUBGROUPS AND QUOTIENT GROUPS 99 

5. If the group of totitives of m has for its order a power of a prime 
this order is of the form 2". 

6. Every p)ossible abelian group is a subgroup of some group of 
totitives. 

Suggestion: If the invariants of the given abelian group are so chosen 
that each is a power of a prime number, it is clearly possible to choose 
m so that the group of totitives of w involves the same invariants. 

40. Subgroups and Quotient Groups of any Abelian Group. 
It has been proved that every abelian group may be regarded 
as the direct product of cyclic groups and hence it is completely 
determined by the orders of these groups. As every subgroup 
of an abehan group is abeUan, it results that these subgroups 
are also completely determined by the orders of the cycUc 
groups of which they are the direct products. Hence it follows 
immediately that a necessary and sufficient condition that an 
abelian group G whose invariants are i\, i2, . . . , i^ contains 
a subgroup whose invariants are ji, J2, , jt is that it be 
possible to associate the tj's with t distinct i's so that each i is 
equal to or greater than the corresponding j . 

If such an arrangement were not possible the subgroup 
would involve more operators of a certain order than the entire 
group. The condition imposed upon the invariants of a sub- 
group is clearly equally applicable as regards the invariants 
of a quotient group. Hence we have the important theorem: 
The invariants of any subgroup of an abelian group are invari- 
ants of a quotient group, ar^d the invariants of any quotient group 
are also invariants of a subgroup. In other words, each subgroup 
is simply isomorphic with a quotient group and vice versa. 
A like theorem is not always true as regards non-abelian groups. 

If a group is cyclic all of its subgroups may be obtained by 
raising successively all of its operators to the same power, but 
this method cannot give all the subgroups of a non-cyclic 
group. The ^th power of each operator of an abelian group G 
gives a group which is simply isomorphic with (7 whenever k 
is prime to g. If k is not prime to g, these ^th powers con- 
stitute a quotient group of G, whose order is g divided by the 
total number of the operators of G whose orders divide k. 



100 



ABELIAN GROUPS 



(Ch. IV 



While it is not diiEcult to find, by means of the theorem 
stated above, the total number of the different types of sub- 
groups in a given abelian group whose invariants are known, it 
is a problem of considerable difficulty to determine all the pos- 
sible subgroups of the same type. To illustrate this fact we 
consider the subgroups of the important class of abelian groups 
of order p^ and of type (1,1, . . . , to m units). In this case 
there are evidently m \ different types of subgroups, exclud- 
ing the identity. That is, there is one and only one type of 
subgroups of order />, a = l, 2, . . . , w 1, separately. 

In this case it is also not very difficult to determine the 
number of the different subgroups of order />". In fact, this 
number is clearly equal to the quotient obtained by dividing 
the total number of ways in which generating operators of such 
a subgroup can be selected from the operators of the group 
by the number of ways in which such generators can be selected 
from the operators of the subgroup. Hence this number is 



{r-^)ir-p)if^-f) 



ir-p"-') 



(P^-iKP^-PHP^-P) 



. ip^-p"-') 
(r-i)(r-^-i) 



(r 



-a+l 



1) 



(/^-l)(/>--l) 



(P-I) 



In the particular case when a = fn l, this formula reduces to 



p-1' 

Hence there are as many subgroups of order />""' in an abelian 
group of order p"* and of type (1,1, 1, . . . ), as there are sub- 
groups of order p. For instance, the group of order 8 and of 
ty-pe (1, 1, 1) has seven subgroups of order 2 and also seven 
subgroups of order 4, while the group of order 16 and of type 
(1, 1, 1, 1) has fifteen subgroups of order 2 and fifteen subgroups 
of order 8, 



4ll ISOMORPHISMS OF AN ABELIAN GROUP 101 

EXERCISES 

1. A necessary and sufficient condition that a group be abelian is that 
each operator corresponds to its inverse in one of the possible auto- 
morphisms of the group.* 

2. Find the number of subgroups with invariants 6, 2 in the abelian 
group whose invariants are 12, 6, 2. 

3. Determine the number of the subgroups of each p)ossibIe order 
in all the abelian groups of order />', p being a prime. 

4. Every abelian group is generated by its operators of highest order. 

5. Give an instance of a non-abehan group which is not generated by 
its operators of highest order. . 

41. Group of Isomorphisms of an Abelian Group, f Some 
of the most useful properties of an abelian group are exhibited 
by its group of isomorphisms. We have already considered the 
group of isomorphisms of a cyclic group and found that it is 
an abelian group. We shall see that a necessary and sufficient 
condition that the group of isomorphisms of an abelian group 
G be abelian is that G be cyclic, and hence it results that the 
groups of totitives are the only abelian groups of isomorphisms 
of abelian groups. This fact is a special case of the theorem 
that the invariant operators of the group of isomorphisms of any 
abelian group constitute a group which is simply isomorphic 
with the group of the totitives of^the largest possible invariant 
of this abelian group. For instance, if an abelian group has the 
invariants 10, 10, 2, the invariant operators of its group of 
isomorphisms constitute the cyclic group of order 4. We pro- 
ceed to prove the stated theorem. 

We shall first prove that if an operator / of the group of 
isomorphisms of an abelian group G transforms every operator 
of G into the same power (7th) of itself it must be commutative 
with every operator of the group of isomorphisms of G. Let 
ti be any other operator of this group of isomorphisms and sup- 
pose that 

This includes the theorem that every group which invdves no operator 
whose order exceeds 2 is abelian. 

t Cf. A. Ranum, Transactions of the American Mathematical Society, voL 
8 (1907), p. 83. 



102 ABELIAN GROUPS [Ch. IV 

Sa being an arbitrary operator of G. Since 

it results that / and h are commutative. On the other hand, 
suppose that / is commutative with every operator in the 
group of isomorphisms of G. We shall first prove that / must 
transform every operator of highest order in G into a power of 
itself. For, if Sa is such an operator and 

where 5a is not a power of Sa, it is clearly possible to find an 
operator /i in the group of isomorphisms of G such that h is 
commutative with 5 but not with s^. As it is necessary then 
that 

it results that / is not commutative with every operator of the 
group of isomorphisms of G unless it transforms every operator 
of highest order of G into a power of itself. 

We shall now show that t must transform into the same 
power every operator of highest order in G, and hence it must 
transform every operator of G into this power, since these opera- 
tors of highest order generate G. It results from the manner 
in which the invariants of any abelian group were determined 
that the group of isomorphisms of G transforms its operators 
of highest order transitively. That' is, the group of isomorph- 
isms of G may be represented as a transitive substitution 
group in which each letter stands for an operator of highest 
order in G. If Sa, s^ represent two operators of highest order 
in G and if 

we can find an operator tz in the group of isomorphisms of G 
such that 

Hence it results that 

That is, /, /2 are not commutative unless 7 = 5. This com- 
pletes a proof of the theorem: A necessary and sufficient con- 



41] ISOMORPHISMS OF AN ABELIAN GROUP 103 

dition that an operator of the group of isomorphisms of an abelian 
group be invariant under this group, is that it should transform 
every operator of this abelian group into the same power of itself* 

Since two abelian groups having the same invariants can be 
made simply isomorphic, and two simply isomorphic groups 
have the same invariants, it results that the order of the group 
of isomorphisms expresses also the number of different ways 
of choosing the independent generators of the group. It 
should be observed that while every operator of highest order 
in an abelian group may be used as an independent generator, 
and hence each operator of highest order must correspond to 
every other operator of this order in some simple isomorphism 
of the group with itself, it is not generally true that every 
operator of lower order corresponds to every operator of its 
own order in some simple isomorphism of the group. 

This fact may be illustrated by means of the abelian group 
whose invariants are p^ and p. It is evident that this group 
contains a characteristic subgroup of order p; viz., the subgroup 
of order p which is generated by its operators of order p^. 
The remaining p subgroups of order p in the given group of 
order p^ are conjugate under the group of isomorphisms of this 
group. 

In any automorphism of any abeUan group G each operator 
of G corresponds to itself multiplied by some operator of G. 
The totality of these multiplying operators evidently consti- 
tutes a group T which is either G itself or a subgroup of G, 
and the automorphism may be obtained by making G isomorphic 
with T and multiplying corresponding operators. In this 
isomorphism no operator except the identity can correspond 
to its inverse. As this condition is necessary as well as sufficient 
we have arrived at the following fundamental 

Theorem: Every automorphism of an abelian group G may 
be obtained by (1) making G isomorphic with one of its subgroups 
or with itself in such a manner that no operator besides the identity 
corresponds to its inverse, and (2) making each operator of G 

* Transactions of the American Maihematical Society, vol. 1 (1900), p. 397; 
vol. 2 (1901), p. 2G0. 



104 ABELIAN GROUPS [Ch. IV 

correspond to itself multiplied by the operator which corresponds 
to it in this isomorphism. 

The simplest case that can present itself is the one in which 
the subgroup of G which corresponds to the identity of T 
includes T. The resulting simple isomorphism of G with itself 
must correspond to an operator in the group of isomorphisms 
of G, whose order is equal to the order of the operator of highest 
order in T. When the order of T is an odd prime number p, 
or double such a number, only one other case can present 
itself; viz., the case in which T, or its subgroup of odd order, 
corresponds to itself in the given isomorphism between G and T. 
In this case the isomorphism corresponds to an operator whose 
order divides p \, in the group of isomorphisms of G. These 
results give rise to the following theorem: // we make an abelian 
group G simply isomorphic with itself by multiplying its opera- 
tors by those of a subgroup whose order is p, or 2p (p being an odd 
prime), the resulting automorphism of G corresponds to an opera- 
tor of order p, 2p, or {p l)/a {a being a divisor of p I), in the 
group of isomorphisms of G. 

The determination of all the possible orders of the corre- 
sponding operators in the group of isomorphisms of any abelian 
group, when T is a given subgroup, seems to be a problem of 
considerable difficulty. When the order of T is small the num- 
ber of cases that have to be considered is also small. In addi- 
tion to the orders included in the given theorems, we have the 
following, when the order of T does not exceed 8: If T is the 
cyclic group of order four, the resulting isomorphism may also 
correspond to an operator of order two in the group of isomor- 
phisms, and when T is the non-cyclic group of this order, it may 
also correspond to operators of orders 3 and 4. When T is the 
cyclic group of order 8, the orders of these operators may be 2, 4, 
and 8; when T is the direct product of the cyclic group of order 
4 and an operator of order 2, the orders of the corresponding 
operators in the group of isomorphisms may be 2 and 4; finally, 
when T is the direct product of three operators of order 2, the 
given operators may be of orders 2, 3, 4, 6, and 7. While all 
of the possible cases for a given T may present themselves in 



42] ISOMORPHISMS OF THE GROUPS OF ORDER p^ 105 

the same group, it is evident that this does not always 
happen. 

For the sake of illustration we consider the group of iso- 
morphism of the group of order 8 which is the direct product 
of three operators of order 2* Each of its 7 subgroups of 
order 4 leads to three operators of order 2. We thus obtain 
the 21 operators of order 2 of the required group of isomor- 
phisms when we consider all the possible instances in which 
the order of T is 2. If the order of T is 4, two cases present 
themselves, in one case just two of the operators of T (includ- 
ing identity) correspond to operators of T, and in the other 
case each one of the operators of T corresponds to some opera- 
tor of T. The former case leads to the 42 operators of order 
4, and the latter to the 56 operators of order 3 of the required 
group of isomorphisms. Finally, we obtain 48 operators of order 
7 when we consider all the possible instances in which the order 
of T is 8. Hence the group of isomorphisms in question is the 
well-known group of degree 7 and of order 168. 

42. Groups of Isomorphisms of the Groups of Order p^. 
A group of order p^ is abelian and there are two such groups. 
The group of isomorphisms / of the cyclic group of order p^ 
is the cyclic group of order p{p \). If 5 is a generator of this 
group of order p^, we may select for a generator of its / any 
operator / of order p{p l) such that t-^st = s", where a is any 
primitive root of p^. If the order of the multiplying subgroup 
T in such an isomorphism is p, the corresponding operators of / 
constitute its subgroup of order p. The remaining operators of 
I result when T is of order p^, p being an odd prime number. 

When the group G of order p^ is non-cycUc, the order of its 
/ is clearly {p^l){p^ p), and this / is isomorphic with a tran- 
sitive group of degree p-\-l corresponding to the transformations 
of the />-(- 1 subgroups of order p contained in G. There is clearly 
a (^ 1, 1) isomorphism between / and this transitive group of 
degree p-\-l, since the p l invariant operators of / are the only 
ones which transform every subgroup of G into itself. 

* C{. E. H. Moore, Bulletin of the A merican Mathematical Society, vol. 1 
(1894), p. 63. 



106 ABELIAN GROUPS [Ch. I\ 

If we regard / as a substitution group on p^\ letters it is 
transitive, its subgroup composed of all its substitutions which 
omit one letter omits p \ letters, and it is a regular group 
on the remaining p^ p letters. This subgroup contains a 
single Sylow subgroup of order p, which corresponds to the 
cases when the multiplying subgroup T is composed of the 
invariant operators of G under the isomorphisms in question. 
The orders of all its other operators divide /> 1 as they corre- 
spond to the cases when T, under the isomorphism in question, 
is a non-invariant subgroup of order p; and this subgroup of 
order p{p l) is simply isomorphic with the metacyclic group 
of degree p, which represents the transformations of p Sylow 
subgroups of G under this subgroup of /. As a substitution 
group on p^l letters the group / is clearly imprimitive when 
p is odd, and its />-}-l systems of imprimitivity are permuted 
according to a doubly transitive group of degree />+l and of 
order p{p^ l), discovered by Mathieu.* 

It is easy to determine the number and the orders of all 
the substitutions of / whose degrees are less than p^ 1. In 
fact, since the subgroup of 7 which is composed of all its sub- 
stitutions omitting one letter is regular, and of order p'^p, 
it follows directly that I involves {p'^p\){p-\-\) different 
substitutions each of which omits exactly p\ letters. The 
order oi pf^ \ of these is p, while the orders of all the others 
are divisors of /> 1. li d represents any divisor of p \, then 
the number of these substitutions which are of order d is 
p{p-\-\)<i>{d). All the substitutions of order p are conjugate, 
but there are <t>{d) equal sets of conjugate substitutions of 
order d and of degree p'^p. The isomorphic group of degree 
/>+ 1 and of order p{p^ l), according to which the p-^1 sub- 
groups of G are transformed, contains also p^1 conjugate 
substitutions of order p, and all its substitutions whose degree 
is less than p are of degree pl and have orders which divide 
pl. The number of these substitutions whose order is d 
is hP(p-\-l)<t>{d). 

E. Mathieu, Paris Comptes Rendus, vol. 47 (IfViS), p. 698. 



43] CONFORMAL GROUPS " 107 

43. Abelian Groups which are Conformal with Non-abelian 
Groups. Two distinct groups are said to be conformal when 
they contain the same number of operators of each order.* 
We proceed to determine all the abelian groups which are con- 
formal with non-abelian groups. The complete solution of 
the converse of this problem, viz., the determination of all the 
non-abelian groups which are conformal with abelian ones, is 
much more difficult, since a large number of distinct non- 
abelian groups may be conformal with the same abelian group, 
while no more than one abelian group can be conformal with 
one non-abelian group. In fact, it was observed in 37 that 
two distinct abelian groups cannot be conformal. 

It is evident that there is only one group of order 2"* which 
does not include any operator of order 4, viz., the abelian group 
of type (1, 1, 1, ,..). Moreover, there is only one cyclic 
group of order 2", and when w<4 no two groups of order 2^ 
are conformal. We proceed to prove that every abelian group 
G of order 2"* which does not satisfy one of these conditions is 
conformal with at least one non-abelian group. 

Let H be the subgroup of G which is generated by the square 
of one of its independent generators s of lowest order, together 
with all the other independent generators of G. The order of 
H is 2"*"^ Since w>3 there is an operator / of order 2 which 
has the following properties: It transforms H into itself, 
it is commutative with half of the operators of 11 (including 
all those which are not of highest order), and it transforms the 
rest into themselves multiplied by an operator of order 2 which 
is not the square of a non-invariant operator of //; i.e., / does 
not transform an operator of order 4 contained in // into its 
inverse. The non-abelian group generated by H and / is con- 
formal with G whenever s- = \. 

When the order of s exceeds two, the group generated 
by / and H (written as a regular substitution group) may be 
made simply isomorphic with itself by writing it on two dis- 
tinct sets of letters. If in this intransitive group / is replaced 
by the continued product of /, the substitution of order two 
Quarterly Journal of Mathematics, vol. 28 (1896), p. 270. 



108 ABELIAN GROUPS [Ch. IV 

which merely permutes corresponding letters of the two sys- 
tems of intransitivity, and s^ in one of the systems of letters, 
there results a transitive group which is conformal with G. 
That is, any abelian group of order 2, m>S, which is neither 
cyclic nor of type (1, 1, 1, ...), is conformal with at least 
one non-abelian group. 

It will now be assumed that the order of G is p"* {p being an 
odd prime number and w>3), and that G is non-cyclic. Let 
H be the subgroup generated by s^ {s being one of the indepen- 
dent generators of lowest order in G) together with all the other 
independent generators of G. There is an operator / of order 
P which transforms H into itself, is commutative with each 
of its operators contained in a subgroup of order />'""^, and 
transforms the rest into themselves multiplied by invariant 
operators of order p. This t and // generate a group which is 
conformal with G whenever 5" = I; for, if si is any substitution 
of H that is not commutative with /, it is easy to see that 

{tsiY = tSitS\ . . . (/> times) 

= tsit-Hhit-Hhit-'^i* . . . /^-''/''5i=5i''.* 

When 5" differs from identity the group generated by H and 
/, written as a regular group, may be made simply isomorphic 
with itself p 1 times, by writing each substitution in p dis- 
tinct sets of letters; and / may be replaced by the continued 
product of /, the substitution of order p which merely permutes 
the corresponding letters of these systems of intransitivity, 
and the pth power of s in one of these systems. In the result- 
ing group the pth power of the operators will be the same as those 
of G taken in the same order, and hence this group will be con- 
formal with G. 

If a non-abelian group whose order is not some power of a 
prime is conformal with an abelian group G, it must be the direct 
product of its Sylow subgroups, and hence each of these sub- 
groups is conformal with an abelian group, and at least onei 
of them is non-abelian. From what precedes, it may be observed 
that necessary and sufficient conditions that any abelian group 

* Transactions of the American Mathematical Society, vol. 2 (1901), p. 262. 



44] CHARACTERISTIC SUBGROUPS 109 

of order Q^'pi'^po"* . . . {pi, p2, . . . being distinct odd 
primes) be conformal with at least one non-abelian group are: 
1^ at least one of its subgroups of orders 2, px\ P'f^ ... is 
non-cyclic; 2^ if the order p^& of this subgroup is odd, then 
a^> 2; if the order is eVen (2), then the subgroup must involve 
operators of order 4 and ao>3. Since any number of these 
factors may be non-abelian, there cannot be an upper limit 
to the number of non-abelian groups which can be conformal 
with an arbitrary abelian group. This fact may be seen in 
many other ways. 

EXERCISES 

1. Let s be of order 16 and let / represent an operator of order 2 such 
that tst=s^\ prove that (5, /) is conformal with the abelian group whose 
invariants are 16, 2. 

2. Find the group of isomorphisms of the abelian group whose invari- 
ants are 8, 2 and determine its invariant operators. 

3. If p is an odd number, there is at least one non-abelian group of 
order p^^ m>2, which is conformal with the abelian group of type (1, 
1, ... to 7M units). The number of such possible groups increases with 
m and has no upper limit. 

4. Any operator of order />* in any abelian group whatever can be 
used as an independent generator provided its />*~^th power is not 
included in a cyclic subgroup of order />*"^^. 

5. Every jxjssible group of finite order is a subgroup of the group of 
isomorphisms of an abelian group of order 2"* and of type (1,1,1,...). 

Suggestions: Observe that this group of isomorphisms contains a 
subgroup which is simply isomorphic with the symmetric group of degree m. 

44. Characteristic Subgroups of an Abelian Group. In 29 
a characteristic subgroup was defined as a subgroup which cor- 
responds to itself in every possible automorphism of the group. 
An operator which corresponds to itself in every possible 
automorphism of the group is Ukewise called a characteristic 
operator. It is clear that every characteristic subgroup is also 
an invariant subgroup, and that every characteristic operator 
is also an invariant operator; but invariant subgroups and 
invariant operators are not necessarily characteristic. The 
Sylow subgroups of an abelian group whose order is not a 
power of a single prime are evidently characteristic subgroups. 



110 ABELIAN GROUPS [Cn. IV 

If G is an abelian group of order ff* and of type (1,1,1, . . . ), 
it contains no characteristic subgroup besides the identity; 
but every other abelian group contains at least one character- 
istic subgroup besides the identity. Suppose that G is an abelian 
group of order />" but not of type (1, 1, 1, ... ), and let />* 
be one of its largest invariants. If exactly Xi of the invariants 
of G are equal to />"', then G contains a characteristic subgroup 
of order p^' and of type (1, 1, 1, ...). This characteristic 
subgroup Ci has been called the fundamental characteristic 
subgroup * of G, since it is contained in every possible char- 
acteristic subgroup of G besides the identity, as we shall prove 
in the following paragraph. 

It is evident that the subgroup of order p^\ which is com- 
posed of the identity and of all the operators of order p which 
are generated by the operators of highest order contained in G, 
is the characteristic subgroup Ci. Moreover, the conjugates 
under the group of isomorphisms / of every operator of order p, 
which is found in G but not in Ci, generate a characteristic 
subgroup of G which includes Ci as a subgroup. This fact 
results immediately from the different possible ways of select- 
ing the independent generators of G. If the second largest 
invariants of G are p"*, and if there are exactly X2 such invari- 
ants in G, then G contains also a characteristic subgroup C2 
of order />^'+^, which is composed of the identity and of the 
operators of order p , which are generated by the operators! 
of order p* contained in G. By continuing this process w 
clearly arrive at the following 

Theorem. // a group G of order />" has Xi invariants whicH] 
are equal to p', X2 which are equal to />"*, . . . , X^ which are\ 
equal to pfi, where a\>a2> . . . >ap, then G has ^ character-l 
istic subgroups Ci, C2, , C^, besides the identity, such thati 
each of them is generated by operators of order p. Their orders] 
are p^\ />^'+^, , p^'+^+ +H, respectively, and each is\ 
included in all of those which follow it. 

Since every operator of highest order contained in an abelianj 
group can be used as an independent generator of the group,] 

* American Journal of Mathematics, vol. 27 (1!K).')), p. 15. 



441 CHARACTERISTIC SUBGROUPS HI 

it is clear that a characteristic subgroup of G cannot involve 
any of its operators of order />"'. All the operators of G whose 
orders divide p'~^', where /3i has any value from 1 to ai 1, 
constitute a characteristic subgroup of G. The character- 
istic subgroup Cff of the preceding theorem corresponds to the 
case when /3i=ai 1. If all the invariants of G are equal, 
there is only one characteristic subgroup of G, besides the 
identity, which involves operators of order p^^ but none of 
higher order. It is also evident that the p^'th. powers of all 
the operators of G constitute a characteristic subgroup of G. 

If 5 is any independent generator of G, the conjugates of 
5 under the group of isomorphisms of G generate a group which 
involves all the operators of G whose orders do not exceed 
the order of s. In other words, if a characteristic subgroup in- 
volves an independent generator of an ahelian group, it also 
involves all the operators of this ahelian group whose orders divide 
the order of this independent generator. This theorem clearly 
includes the theorem stated above, to the effect that a char- 
acteristic subgroup cannot involve any of the operators of 
highest order contained in G. In the following paragraph we 
shall estabUsh a still more general theorem in case />> 2. 

For a study of the special properties of the characteristic 
subgroups it is convenient to let Hi, H2, . . . , H^ repre- 
sent the subgroups of G which are generated respectively by 
a set of Xi independent generators of order />", a set of X2 
independent generators of order p^, . . . , a set of X^ indepen- 
dent generators of order p". Suppose that p>2 and that s\ 
is some operator of order />*, a\> 5, which is contained in G, 
If ay> b^Uyj^i, and if S\ is the product of an operator of highest 
order in Hy+i and an operator of order p^ from Hy, then the 
conjugates of S\ under / generate a group which involves all 
the operators of order />* that are contained in G. In fact, 
this group clearly involves an independent generator of Hy^x 
since p>2, and it involves all the operators of order p* in the 
direct product of the subgroups Hi, ... , Hy. 

By means of the preceding theorems it is not difficult to 
determine the characteristic subgroups of any given abelian 



112 ABELIAN GROUPS [Cm IV 

group of order p^. *If the order of an abelian group is not a 
power of a single prime number its characteristic subgroups 
are found by forming the direct product of the characteristic 
subgroups of its Sylow subgroups, and all such direct products 
are characteristic subgroups of G. 

EXERCISES 

1. If an abelian group G of order />"" has only two distinct invariants 
pcn^ pat^ and if aiai = n, then the number of the characteristic subgroups 
which are generated by the operators of order p^ is 2, when =1 and h 
has any one of the values from 1 to ai 1, The number of these subgroups 
cannot exceed the smaller of the two numbers M-f 1, aj + 1 for any value 
of . 

2. Find all the characteristic subgroups of the abelian group of order 
/ and of type (1, 2, 3). 

3. The abelian group of order 16 and of type (1, 3) has the propert\- 
that no single set of operators of order 4, which are conjugate under its 
group of isomorphisms, generates all its operators of this order. Prove that 
whenever p>2 all the operators of order p'^ in the abelian group of order 
p* and of type (1, 3) are generated by a single set of operators of order p- 
which are conjugate under its /. 

45. Non-abelian Groups in which Every Subgroup is Abelian. 

Let G represent any non-abelian group all of whose subgroups 
are abelian. As instances of such groups we may cite the 
octic and quaternion groups. We shall first prove that G 
must contain an invariant subgroup of prime index p. Suppose 
that G is represented as a transitive substitution group of the 
smallest possible degree. If this group is imprimitive it must 
transform a set of systems of imprimitivity according to a prim- 
itive group which has a (1, a) isomorphism with G. 

This primitive group must be such that each of its subgroups 
is abelian, and hence we have only to prove that every prim- 
itive group which contains only abelian subgroups has an 
invariant subgroup of index p. If this primitive group were 
regular it would be of order p. If it were non-regular and of 
degree , a maximal subgroup of degree w 1 would be abelia 
and hence all of its substitutions besides the identity would 
of degree 1 ; for, if the degree of such a substitution were 
less than w 1 , this substitution would occur in two maximal i 



45] GROUPS CONTAINING ONLY ABELIANSUBGROUPS 113 

abelian subgroups and hence it would be invariant under the 
entire group. Consequently this primitive group of degree n 
would involve exactly 1 substitutions of degree n, while 
each of its remaining substitutions besides the identity would 
be of degree 1. 

The substitution groups which have these properties have 
been studied extensively. Frobenius * proved that such a 
group must have an invariant subgroup of order n. This im- 
portant fact will be proved in 139. If we assume this theorem 
for the present, it results that G must contain an invariant 
subgroup of index p, since the quotient group with respect to 
the given subgroup of order n is abelian. 

We shall now prove that the order g oi G cannot be divisible 
by more than two distinct prime numbers. Supp>ose that 
g = pj^p2* . . . px\, where />i, />2, . . . , /x are distinct prime 
numbers. Since G contains an invariant abelian subgroup of 
prime index, we may suppose that it contains an invariant 
subgroup H of order h, where h is given by the formula 

A = />i"'-V2"' . . . px"^- 

As H is the direct product of its Sylow subgroups and as every 
operator of G which is not in H has an order which is divisible 
by pi, it results directly that G contains only one Sylow sub- 
group of each of the orders p2% , p\">'- 

Let s be any operator, which is found in G but not in H 
and whose order is of the form //. As 5 transforms each of 
the Sylow subgroups of H into itself, and as G is non-abelian, 
J must be non-commutative with some of the operators in one 
of these Sylow subgroups. This Sylow subgroup and s must 
generate G, otherwise G would involve a non-abehan subgroup. 
This completes the proof of the fact that the order of a non- 
abelian group which contains only abelian subgroups cannot be 
divisible by more than two distinct prime numbers. 

Suppose that the order of G is />i"/>2"', a2>0, and that G 
contains an invariant subgroup of order />i'"*/>2"', and represent 
the Sylow subgroups of orders pi^ and p-z"* by Pi and P2 respec- 

* Frobenius, Berliner SUzungsberickte, 1902, p. 455. 



b 



114 ABELIAN GROUPS [Ch. IV 

tively. We proceed to prove that Pi is cyclic and that P2 is of 
type (1,1, 1, ...). That Pi is cyclic follows directly from 
the fact that s transforms P2 into itself and that G is generated 
by s and P2. In fact, it is evident that ai=/3. If P2 were 
not of type (1, 1, 1, . . . )> it would contain characteristic 
subgroups generated by its operators whose orders are divi- 
sors of />2, p2^, . . , p2^~^, where P2'' is the order of its opera- 
tors of highest order. All of the operators of these character- 
istic subgroups would be composed of operators which would 
be commutative with s, since G cannot contain a non-abelian 
subgroup. Hence s would have to transform among them- 
selves all the operators of order P2'' in P2 which have the same 
p2th power. As the number of these operators is a power 
of p2, this is impossible. That is, we have arrived at an absurd- 
ity by assuming that r> 1, and hence we have established the 
theorem: If a non-abelian group which contains only ahelian 
subgroups has more than one Sylow subgroup, one of these sub- 
groups is of the type (1, 1, 1, ... ) and the others are cyclic. 

EXERCISES 

1. If all the subgroups of a non-abelian group of order />*", p being a 
prime number, are abelian, its commutator subgroup is of order p and 
the pih power of each of its operators is invariant.* 

2. Every subgroup of the dicyclic group of order Ap is abelian. 

3. If every subgroup of order p"^~^ in a non-abelian group of order 
/>"* is abelian, there must be exactly p-\-\ such subgroups. 

4. The group of order 56 which contains 8 subgroups of order 7 does 
not involve any non-abelian subgroup. 

46. Roots of the Operators of an Abelian Group. If 5i, 52, 

. . . , Sg are the operators of an abelian group G, and if G con- 
tains two operators 5, s^ which are such that Sa=Sfi, then s^ 
is said to be an th root of s^. In particular, every operator 
of G is a gth root of the identity, so that the identity has g gth 
roots under G. If n is prime to g every operator of G has one 
and only one nth root. On the other hand, if w is a divisor 
of g, the total number of the operators of G whose orders divide 

* Cf. G. A. Miller and H. C. Moreno, Transactions of the American Matlte- 
matical Society, vol. 4 (1903), p. 403. 



47) HAMILTON GROUPS 116 

n is always divisible by n. These operators constitute a sub- 
group III of G. If an operator of G has one th root it must 
have //I such roots, hi being the order of Hi. 

Whenever n is a divisor of g all the operators of G may be 
di\'ided into two classes according as they have nth roots or do 
not have this property. As each operator of the first class has 
exactly hi nth roots, the number of the operators in this class is 
g/hi. The number of the operators in the second class is there- 
fore g-g/hi. When G is cyclic, hi=n. 

If d is the highest common factor of n and g every operator 
of G which has one th root must have either d or a, multiple 
of d such roots, since the total number of the operators of G 
whose orders di\dde d is divisible by d. The number of the 
operators which have th roots is evidently equal to g/h2, A2 
being the order of the subgroup H2 which is composed of all the 
operators of G whose orders divide d. 

The h2 operators of G which are th roots of the same 
operator correspond to the same operator in the quotient 
group G/H2. Every operator of G is clearly an wth root of one 
and of only one operator of G, but a given operator may have 
a number of different th roots. If a group is non-abelian two 
operators which have th roots need not have the same number 
of such roots. 

For instance, in the symmetric group of order 6 the identity 
has three square roots while each of the two operators of order 
3 has only one such root. 

47. Hamilton Groups. A non-abelian group G is said to be a 
Hamilton group, or a Hamiltonian group, if each of its subgroups 
is invariant. WTiile these groups are not abelian, they have 
in common with the abelian groups the property that every 
subgroup is invariant. They were thus named by Dedekind 
in honor of Sir W. R. Hamilton, and some of their fundamental 
properties were first studied by Dedekind in view of their 
usefulness in the study of the number reahns which belong to 
such groups.* 

Since every subgroup of G is invariant, G must be the direct 

* R. Dedekind, Mathematische Annalcn, vol, 48 (1897), p. 548. 



116 ABELIAN GllOUPS (Ch. IV 

product of its Sylow subgroups. // all the subgroups of a Sylow 
subgroup of odd order are invariant, this Sylow subgroup must 
be abelian. In fact, if the order of such a subgroup Pm is />*", 
then each of its operators of order p must be invariant, since the 
group of isomorphisms of the cyclic group of order p is of order 
/> 1. Suppose that Pm contains a cyclic subgroup C^ of order 
P" such that Ca involves non-invariant operators. 

If this were possible Pm would contain an operator ^ which 
would transform C into itself without being commutative with 
every operator of Ca. As the operators of Ca would also trans- 
form into itself the cyclic group Sp generated by 5, it follows 
that the commutators, involving elements from Ca and Sp, 
would be in both of these cyclic groups. These commutators 
would therefore be in the central of the group generated by 
Ca and Sp. We may suppose that s was so selected that s^ is 
commutative with every operator of Ca- As the group gener- 
ated by Ca and this 5 would contain non-invariant operators 
which would not generate the commutator subgroup of order 
p, it results that Pm must be abelian. 

We have now proved that every Hamilton group is the direct 
product of a Hamilton group of order 2"* and some abeUan group 
of odd order. Hence it remains only to determine the possible 
Hamilton groups of order 2*". As an instance of such a group 
we may cite the quaternion group. We shall first prove that 
such a group H cannot involve any operator whose order ex- 
ceeds 4. 

In fact, if H contained an operator s whose order exceeds 4, 
we could find a subgroup in H, by the method used above, 
which would contain a non-invariant operator which would 
not generate the commutator of this subgroup. Hence we can 
assume that the order of every operator of H is either 2 or 4. 
Moreover, the operators of order 2 are contained in the central 
of H, while each operator of order 4 is transformed either into 
itself or into its inverse by every operator of H. Two of 
these non-commutative operators of order 4 must have a com- 
mon square, and hence any two such operators must generate 
the quaternion group. 



47) HAMILTON GROUPS 117 

The group generated by any one such quaternion group and 
the operators of order 2 contained in II must coincide with //. 
In fact, if an operator s oi H were not contained in this group 
this operator and this group would generate a group which 
would involve more operators of order 2. As this is contrary 
to the h>T)othesis, we have established the following theorem: 
Every possible Hamilton group is the direct product of a quaternion 
group, an ahelian group of order 2"* and of type (1, 1, 1, . . .), 
and an abeliati group of odd order * 

A group which is a direct product of two groups is some- 
times called a divisible group. If it is not a direct product 
it is said to be indivisible. Hence the quaternion group is the 
only indivisible Hamiltonian group. The only indivisible abelian 
groups are the cyclic groups whose orders are powers of prime 
numbers. If an abelian group is written as the product of indi- 
visible groups, the orders of these groups constitute the largest 
possible set of invariants of the abelian group. 

EXERCISES 

1. The commutator quotient group of a Hamilton group of order 2* 
is abelian and of type (1, 1,1,...). 

2. The number of the possible Hamilton groups of order 2";fe, k being 
any odd number, is equal to the number of the abelian groups of order k. 

3. Every Hamilton group has the four-group as a group of inner iso- 
morphisms. 

4. Two arid only two of the operators of a Hamilton group are character- 
istic. 

5. Let g=Pi"Pt" . . . ^x"'^, where pu Pt, . . ,P\ are distinct primes. 
Necessary and sufficient conditions that all existing groups of order g 
shall be abelian are: (1) each otj^2; (2) no p"J l is divisible by one 
of the primes />i, p>, . . , P\. Cf. L. E. Dickson, Transactions of the 
American Mathematical Society, vol. 6 (1905), p. 201. 

* This theorem, together with various other theorems relating to Hamilton 
groups, was proved by G. A. Miller, Bulletin of the American Mathematical 
Society, vol. 4 (1898), p. 510. Some cf these theorems were proved several years 
later by E. Wendt, Mathematische Annalen, vol. 59 (1904), p. 187. In the fol- 
lowing volume of this journal Wendt corrected this oversight. 



CHAPTER V 
GROUPS WHOSE ORDERS ARE POWERS OF PRIME NUMBERS 

48. Introduction. It has been observed in 11 that if 
/>" is the highest power of the prime p which divides the order 
of a group G then G must involve at least one subgroup of order 
p^, and if G involves more than one such subgroup, all the sub- 
groups of this order (Sylow subgroups) form a complete set of 
conjugates. These facts indicate that it is especially important 
to know the fundamental properties of Syloiv's groups; * that 
is, of groups whose orders are powers of prime numbers. For- 
tunately all these Sylow groups have unusually interesting 
properties in common and they offer more easy avenues of 
penetration than the groups whose orders are arbitrary num- 
bers. 

A strong instrument of attack here, as well as in many other 
places in group theory, is the concept of complete sets of con- 
jugates. Each non-invariant operator of a non-abelian group 
G of order p"* belongs to a complete set of p conjugates, since 
such an operator is transformed into itself by all the operators 
of a subgroup whose order is p^, /3<w. Hence all the non- 
invariant operators of G occur in sets, such that each set involves 
a power of p conjugate operators, and each non-invariant 
operator occurs in one and in only one set. The total number 
of the non-invariant operators must therefore be of the form 
pk] and, as there are />'" 1 operators besides the identity in 
G, there must be an invariant operator of order p in G. This 

The groups whose orders are powers of prime numbers are also known as 
primary groups. G. Frobenius and L. Stickelberger, Journal reine angav. Math., 
vol. 86 (1879), p. 219. They are sometimes called prime-power groups. In 
view of the unusually large number of useful theorems in this field these groups 
have been said to constitute the El Dorado of the theory of groups. Bulletin oj 
the American Mathemalical Society, vol. 6 (1900), p. 393. 

118 



48] INTRODUCTION TO PRIME-POWER GROUPS 119 

theorem was first proved by L, Sylow * and it may be regarded 
as the most important theorem relating to the prime-power 
groups. 

It has been observed that the totality of the invariant 
operators of any non-abelian group constitutes an important 
subgroup known as the central. With respect to its central, 
G is isomorphic to a group of order />"', m' <m. This quotient 
group must also have a central subgroup, if it is non-abelian, 
and this gives rise to a second quotient group of order />"", 
m" <m' . By continuing this process we must arrive at an 
abehan quotient group. It is a matter of considerable impor- 
tance to observe that this abelian group is never cyclic. In 
fact, this is a special case of the theorem proved in 28 that 
the central quotient group of a non-abeUan group is always 
non-cyclic. 

The subgroup of G which corresponds to an invariant sub- 
group of order p in the central quotient group of G is abelian, 
but includes operators which are not in the central of G. Hence 
it results that eroery non-abelian group of order p^ contains an 
invariant abelian subgroup whose operators are not separately 
invariant under the group. 

Since we can always arrive at the identity by forming 
successive central quotient groups of G it results that G must 
have at least one invariant subgroup whose order is an arbi- 
trary divisor of the order of G. Suppose that ^i, ^2, , 
Hp represent any complete set of conjugate subgroups of G. 
Since each of these subgroups is transformed into itself by a 
subgroup of G whose order is a power of />, it results that p 
is also a power of p. Hence each of these H's must transform 
into itself each one of at least /> l of the other ^'s, since 
it transforms itself into itself, and since it must transform a 
multiple of p of these conjugates among themselves. That 
is, each one of a complete set oj conjugate subgroups of a group of 
order p^ is transformed into itself by at least p \ others of the set, 
if the set includes more than one subgroup. In particular, every 
subgroup of order />""' in a group of order />" is invariant under 

* Malhcmaiische Annalen, vol. .'> (1872), p. 584. 



i20 CRIME-POWER GROUPS [Ch. V 

this group. As has been observed in 28, this theorem is also 
a special case of the following theorem: If Hi and H2 are two 
conjugate subgroups of G, then the index of Hi or H2 under G 
is greater than the index under Hi or H2 of the cross-cut of Hi 
and H2. 

49. Invariant Abelian Subgroups. From the fact that every 
group of order />" contains at least p invariant operators and 
that its central quotient group has the same property, it results 
that every group of order p"*, m>2, contains a subgroup of 
order p^~^ which involves p^ invariant operators. In a similar 
way we observe that every group of order p^, m>5, contains 
a subgroup of order p^-'^-^ which involves p^ invariant opera- 
tors. In general, every group of order />*", w>(a+2)(a 1)/2, 
contains a subgroup of order 

^-1-2-3-,.. -(a-l) _ j,m-o(a-l)/a 

which involves />" invariant operators. As the group formed 
by these invariant operators corresponds to an invariant sub- 
group in the quotient group, it results that it is invariant under 
the . entire group. Since every invariant subgroup of order pf* 
in a group of order p^ is contained in an invariant subgroup of 
order p"'^^, it results from the above that every group of order 
/>", m>a{a+l)/2, contains an invariant abelian group of order 
//*+*. In other words, every group of order />"*, w>/3(/3 1)/2 
contains an invariant abelian subgroup of order p^. 

In the special case when p = 2 this theorem may be expressed 
in a little more general form as follows: 

Every group of order 2*", m f i3(i3- 1)/2, /3> 3, contains an abe- 
lian subgroup of order 2^ *. The proof of this extended theorem 
is short if the preceding developments are employed. In 
fact, it has been proved that G involves a subgroup of order 

pm-l-2- ... -(/3-3) _ Am-(/3-2)J-3)/2 /3>3 

* In fact, there is always an invariant abelian subgroup of order 2^ when the 
given conditions are satisfied. Cf. Messenger of Mathematics, vol. 41 (1912), 
p. 28. 



491 INVARIANT ABELIAN SUBGK0UP8 121 

whose centralis of order at least />^"*, whenever m^(0\) 
(/3- 2)/2. When w = /3(/3- 1)/2 the order of this subgroup is 

pfi(fi-l)/2-l0-2)(0-Z)/2 _ f,20-3 

The quotient group with respect to the given central is of order 
p^~^. If this quotient group contains operators of order ^, 
G must evidently involve an abelian group of order p^. It 
remains therefore to consider the case when this quotient 
group does not involve any operator of order p^. li p = 2 we may 
assume that this quotient group is abelian, and hence we shall 
confine our attention, in what follows, to this special case. 

We are thus led to consider the possibility of constructing 
a group K of order 2^'^, having a central C of order 2^"^ which 
leads to an abelian quotient group of t>pe (1, 1, 1, ...). 
If we arrive at a contradiction by assuming that K does not 
include an abeUan subgroup of order 2^ our theorem is proved. 
If K existed, all the operators of C besides the identity would 
be of order 2, since all of these operators would be commutators 
of K. Moreover, each of the .non-invariant operators of K 
would be transformed under K into itself multiplied by all 
the operators of C. 

Let Ki represent any subgroup of order 2^""* and involving 
C Each of the non-invariant operators of Ki is transformed 
under Ki into itself multiplied by all the operators of a subgroup 
of order 2^"^ contained in C. The multiplying subgroups for 
two distinct operators (mod C) of Ki must be distinct, other- 
wise the operators of the group of order 4 (mod C) generated 
by these two operators would have to be transformed, by an 
operator of K which is not also in Ki, into themselves multi-' 
plied by the operators of a group of order 4 which has only the 
identity in common with the given subgroup of order 2^"' 
in C. As this is clearly impossible it results that all the dif- 
ferent non-invariant operators of Ki are transformed under 
Ki into themselves multiphed by all the different subgroups of 
order 2'*-' in C. 

From the preceding paragraph it results that there is a (1, 1) 
correspondence between the operators of A'l and the subgroups 



122 PRIME-POWER GROUPS (Ch. V 

of order 2^"' in C such that each operator of K\ is transformed 
under K\ into itself multiplied by the various operators of the 
corresponding subgroup. Let h be any non-invariant operator 
of A'l and consider all the possible subgroups of order 4 in the 
quotient group of K\ with respect to C, such that each of these 
subgroups involves the operator corresponding to /i. Any oper- 
ator (p) of K which is not also in K\ transforms each of these 
subgroups into itself multiplied into a subgroup of order 4 con- 
tained in C. Let/i, ^2, . . , //3-2 represent a set of operators 
of K\ which correspond to a set of independent generating 
operators in the given quotient group and assume that 

ti~^S2h=S\t2, h-'^Szh=S2h, , /i"^-y^-2^i=-J/s-3^/j-2- 

The subgroup {h, ti) is transformed by p into itself multiplied 
by a group of order 4 which does not involve s\. In general, 
the subgroup (ii, /), a = 2, 3, . . . , jS 2, is transformed by 
p into itself multipHed by a subgroup of order 4 of C which 
does not involve 5_i, and {h, ta^tat ta^ is transformed 
by p into itself multiplied by a subgroup of order 4 which does 
not involve 5a._i5a,_i . . . 5j^_i, ai, 0:2, . . . ,ax = \,2, . . . , 
/3 2. As p must transform h into itself multiplied by an oper- 
ator which is common to all of these subgroups of order 4 and 
as 5i, 52, . . . , Sp-z are independent generators of a group of 
order 2^"^, it results that pti=tip, which is contrary to the hy- 
pothesis. That is, we have arrived at a contradiction by assum- 
ing that K does not involve an abeUan subgroup of order 2^ 
and hence the theorem under consideration has been proved. 

EXERCISES 

1. In a group of order ^"' the order of the commutator subgroup cannot 
be greater than p"*~^. 

2. In a non-abelian group of order />' each of the non-invariant 
operators belongs to a complete system of p conjugates. 

3. There is one and only one non-abelian group of order />', p>2, which 
is conformal with the abelian group of type (1, 1, 1). 

4. If a non-abelian group of order />*", P>2, contains an operator of 
order p^~ * its commutator subgroup is of order p and there is only one such 
group. 



501 NUMBER OF THEIR SUBGROUPS 123 

5. Every non-abelian group of order ^"* contains an invariant conimuta- 
tor of order p. 

6. If a group of order 3"* contains no operator of order 9 all of its opera- 
tors in any complete set of conjugates are commutative.* 

Suggestion: If 5i, Si are any two operators of such a group it results 
that {sis^^= Si- SiSiSi^ Si'^S\Si= {siSi}y=Si-SiHxSi-StSiS%^= 1. 

7. A necessary and sufficient condition that a group of order ^"* is abe- 
lian is that more than />"*"* of its operators corresponds to their inverses 
in some automorphism of the group. 

8. If two non-commutative operators of a group of order />"*, P>2, 
correspond to their inverses in an automorphism of the group their com- 
mutator cannot correspond to its inverse in this automorphism. 

50. Number of Subgroups in a Group of Order p^. We 

shall first determine the form of the number of subgroups of 
order />"*"^ in a group G of order f^. Any two subgroups 
of order f^~^ must have p^~'^ operators in common, and these 
common operators constitute a group which is invariant under G. 
They must therefore include all the commutators of G and also 
the pth. powers of every operator of G. If H is composed of all 
the operators which are common to all the subgroups of order 
/>"*"^ contained in G it must include all the commutators of 
G as well as the p\h powers of all its operators. From this 
it results that the quotient group corresponding to H is abelian 
and of type (1, 1, 1, ...). Each subgroup of order />"*"* 
in G must correspond to a subgroup of index p in this quotient 
group. In Chapter IV, 40, we proved that the number of these 
subgroups of index p is equal to the number of the subgroups 
of order p in this quotient group. Hence the theorem: 

The number of subgroups of order />""* contained in a group 
G of order />"* is always of the form 

p^-l 
p-\ 

To obtain the exact number of these subgroups it is neces- 
sary to observe that p*" is the order of the quotient group 
of G with respect to the group formed by all of its operators 

W. Bumside, Quarterly Journal of Mathematics, vol. 33 (1901), p. 231. 



124 pr;me-power groups ICh. V 

which are common to all of its subgroups of order T''- I" 
the special case when G is abelian and of type (1, 1, 1, ... ), 
X = w; but in all other cases, m>\. Since in any group every 
subgroup which involves half the operators is invariant, it 
results from the above that the number of subgroups whose 
order is one-haK the order of the group is always of the form 
2_i. In particular, there is no group which involves exactly 
5 or 9 subgroups of half its order. The last two statements 
are evidently not restricted to groups whose orders are powers 
of a single prime. In fact, it results from the given method 
of proof that the number of the invariant subgroups of index 
* in any group whatever which contains at least one such 
invariant subgroup is of the form {p^-l)/{p-l)- This is 
known as Bauer's theorem* 

Having found the number of the subgroups of order p"^ 
in G, we shall now consider the other extreme case and deter- 
mine some property of the number of its subgroups of order 
p. The invariant subgroups of order p which are contained 
in G clearly generate a group of order / which^contains 
(^pP _ !)/(/, - 1) distinct subgroups of order p, where /3 > 1 . The 
non-invariant subgroups of order p may be divided into complete 
sets of conjugates, each set containing p'', 7>0, distinct groups. 
Hence the total number of subgroups of order p in any group 
whose order is a power of p is always of the form l-\-kp. From 
this theorem it results immediately that the number of the sub- 
groups of order />"+' which contain a given subgroup of order 
f* is always of the form l-\-kp. 

We proceed now to consider the number of subgroups of 
order /> in G, where p is an arbitrary divisor of />. We assume 
at first that a<m, and denote by r the number of subgroups 
of order p" in G and by r+i the number of these subgroups 
of order />"+^ Let Sx represent the number of the subgroups 
of order />-+' in which a given subgroup of order p" occurs, 
and let Sy denote the number of subgroups of order p' contained 
in a given subgroup of order r""'- We then count each sub- 

* H. Hilton, Introduction to the Theory of Groups of Finite Order, Oxford, 
1908, p. 145. 



5 501 NUMBER OF THEIR SUBGROUPS 125 

group of order r-. as many times as it contains different 
subgroups of order ^ and tlius arrive at tl,e following equaUon: 



I-l 1,-1 



As both . and .are congruent to unity modulo p it results 
that r =r^,(mod p). As n^l(mod p) it results that r.^1 
(mod p) and hence tite number of the subgroups of order p" in a 
group of order />" is always of the form 1 -{-kp. 

It is now very easy to prove that the number of the sub- 
groups of order r in any group G' whose order is divisible by 
P IS always of the form l^kp, even if the order of G' is not a 
power of p. If a subgroup of order ^ in G' is not invariant 
under some one of the Sylow subgroups of order r in G' it 
evidently belongs to a complete set of ps conjugates. Hence 
we niay confine ourselves to these subgroups of order ^ in G 
wbch are mvariant under a particular subgroup of order r 
m G All of these must occur in this Sylow subgroup of order 
r and hence their number is of the form 1 ^kp. This proves 
tiiat the total number of the subgroups of order /,- must also 
be of this form, so that we have proved the following theorem 
due to Frobenius: 

The total number of the subgroups of order p- in any .roup 
whose order is divisible by p- is of the form 1+kp -^ ^ 

This theorem holds whether the order of the group is or is 
not a power of a single prime, and it may be regarded as an 
exten^n of Sylow 's theorem. It should, however,\e obse'ed 
that the subgroups of order p" are not always conjugate when 
they are not Sylow subgroups. . ^ J h ^ wnen 

If the group G of order ^ contains at least one abelian 
subgroup of order p^ it is easy to prove that the number of Jts 
abehan subgroups of order /.^ is of the form 1 -\-kp. We proceed 
to prove this theorem. If G contains an abelian subgroup 
B of order r the totahty of the operators of G which are com 
mutative with every operator of 77 forms a group H' which 
mcludes aU the abelian groups of order p"^^ that are con 
tamed m G and include H. Hence the abelian subgroups of 



126 PRIME-POWER GROUPS (Ch. V 

order />*"*"* in G which include H correspond to the subgroups 
of order p in the quotient group of H' with respect to E. As 
the number of these subgroups of order p is of the form \-\-kp 
whenever H and H' are not identical, we have proved that an 
abeUan group of order p" contained in G is found in l-\-kp 
abeUan subgroups of order /?""*" ^ whenever it is found in at least 
one abelian subgroup of this order. 

Suppose that G contains at least one abelian subgroup of 
order /j""^^ and let Ta+i represent the number of its abelian 
subgroups of this order, while r represents the number of the 
abelian subgroups of order p" contained in G and included in 
abelian subgroups of order p"^^. By counting each abelian 
subgroup of order />""" ^ as many times as it contains subgroups 
of order />" and denoting by Sx and Sy respectively the number 
of the abelian subgroups of order ^"+^ in which a given subgroup 
of order p" occurs, and the number of subgroups of order />" 
in a given abelian subgroup of order />"*" ^ we obtain, as before, 
the equation 

S 5^= 2 Sy. 

x-i v-y 
Since both Sx and ^i, are congruent to unity modulo p we have 
proved that ra = ra+i (mod p). As ri = l (mod p) and r2^1 
(mod p) whenever G contains an abelian subgroup of order p'\ 
there results the theorem : 

The number of the abelian subgroups of order p^ in any group 
of order p^ is either zero or of the form l-{-kp. 

It should not be inferred that the number' of the abelian 
subgroups of a group of order p^ is always either 0, or of the 
form 1 -\-kp. The number of the abelian subgroups of index p 
in a non-abelian group of order p^ is either 1 or 1 -\-p, if this 
number exceeds zero, according to example 3 of the following 
exercises; but there are non-abelian groups of order />" which 
contain exactly two abelian subgroups of index p-. The pos- 
sible numbers of abeUan subgroups in a non-abelian group of 
order />" have not yet been determined. 

By means of Bauer's theorem and the properties of the 
^subgroups it is easy to prove a fundamental theorem relating 



501 NUMBER OF THEIR SUBGROUPS 127 

to the various possible sets of independent generators of any 
prime power group. As every maximal subgroup of a group 
of order f' is of order p^-\ it results directly from the defini- 
tion, that the 0-subgroup of any group of order f is the cross- 
cut of all its subgroups of index p. Hence the (^-subgroup of 
a group of order f" can also be defined as its smallest invariant 
subgroup which gives rise to an abelian quotient group of type 
(1, 1, 1, . . . ). If the order of this quotient group is p", it 
follows that a is the number of independent generators in 
every possible set of such generators of this group of order />*". 
In particular, every possible set of independent generators of any 
prime power group involves the same number of operators. That is, 
the number of operators in each of the possible sets of indepen- 
dent generators of a Sylow group is an invariant of this group. 
From the preceding paragraph it results directly that a 
necessary and sufficient condition that the <^-subgroup of a 
given group of order p^ be the identity is that this group be the 
. abeUan group of type (1, 1, 1, . . . ). Hence there is one 
and only one group of order />", p being any prime number 
and m being any positive integer, which has the identity for 
its 0-subgroup. The number of operators in every set of inde- 
pendent generators of this group is m. In every other group 
of order />" the number of these independent generators is less 
than w, and there is at least one group of order f in which 
this number is any arbitrary positive integer from 1 to m. 

EXERCISES 

1. The number of abelian subgroups of order /> in any group is either 
or of the form 1 -{-kp, even if the order of the group is not a power of p. 

2. If an abelian group of order p"^ has p'- for one invariant while all 
of its other invariants are equal to p, the number of its subgroups of order 

^-1 is P^l^, 
p-\ 

3. A non-abelian group of order p''^^ contains 0, 1, or p + l abelian sub- 
groups of order />*; in the last case it contains />'"' invariant operators. 

4. If a group of order 2* contains only one subgroup of order 2 it is 
either cyclic or dicyclic, and if a group of order />"*, p>2, contains only 
one subgroup of order p it must be cyclic* 

W. Bumside, Theory of Croups, 1911, p. 132. 



128 PRIME-POWEll GROUPS (Ch. V 

5. If a group G of order />"* contain only one subgroup of order p", 
1 <a<m, the group is cycltc. 

Suggesliott: Every operator of G which is not in the subgroup of 
order p" must generate this subgroup, since every subgroup of order p" 
in a group of order />"* is contained in a subgroup of order />"*'*. 

6. The number of the invariant subgroups of order p" in any group 
of order p'^ is of the form 1 +kp whenever w > . 

7. There are just four non-abelian groups of order 2" involving a 
cyclic group of order 2*""*, m>S. 

8. Prove that the number of the subgroups of order p in any group of 
order p" is of the form 1 +kp, by observing that the number of operators 
of order p" in such a group is of the form kap"~^{p 1), so that 

^m_i^"2\/-^(/.-l).. 

9. Find a group of order 18 which contains only two invariant sub- 
groups of order 3, and hence prove that the number of invariant subgroups 
of order />" in a group need not be either or of the form I -{-kp. 

51. Number of Non-cyclic Subgroups in a Group of Order 
J*"*> i^>2. It has been observed that the total number of the 
subgroups of order />** in any group G of order p^ is of the form 
\-\-kp whenever m>a, and hence the number of the invariant 
subgroups is always of this form. The number of the abelian 
subgroups is again of this form whenever a < 4 and this number 
is greater than zero. We proceed to prove that the number 
of the non-cyclic subgroups is always of the form l-\-kp when- 
ever p>2. This proof can easily be effected by a method 
employed above. It is easy to prove that the number of the 
non-cyclic subgroups of order ^" in a group of order p"'^^ is of 
the form l-\-kp whenever this number is not zero and p>2 
by showing that there must be p cyclic subgroups of order 
/>* whenever there is one such subgroup. The number of the 
subgroups of order />""*" S which contain a given non-cyclic 
group of order p and are themselves contained in a group of 
order p^, is also of the form l+kp. 

Let fa and r^+i represent respectively the numbers of the 
non-cyclic subgroups of order p and />"*"', and let Sx represent 
the number of the subgroups of order />"""' in which a given 
non-cyclic subgroup of order p occurs while Sy denotes the num- 



521 NUMBER OF NON-CYCLIC SUBGROUPS 129 

ber of non-cyclic subgroups of order p" contained in a given 
subgroup of order p""^^. We then count each subgroup of 
order />+^ as many times as it contains a non-cyclic subgroup 
of order />" and thus arrive at the equation 

2j Sx ^ ^ Sy% 

X-l y.l 

Since both Sx and Sy are congruent to unity modulo p, it results 
that fa^Ta+i (mod p) whenever G contains at least one non- 
cyclic group of order p" and m>a. Since every non-cyclic 
group of order p^ contains at least one invariant non-cyclic 
group of order p^ whenever p>2, there results the theorem: 

The number of the non-cyclic subgroups of order p" in any 
non-cyclic group of order p^ is of the form 1+kp whenever l<a<m 
andp>2.* 

52. Number of Non-cyclic Subgroups in a Group of Order 2"*. 
It is known that there are three non-cycUc groups of order 2"* 
which contain a single cyclic subgroup of order 2', 2<a<m; 
viz., the three groups of order 2"* which involve a cyclic subgroup 
of order 2*""^ and transform a generator of this subgroup into 
its inverse or into its (2"*~^ l)th power. It is not difficult to 
prove that these are the only possible non-cyclic groups of order 
:'^ that contain an odd number of cyclic subgroups of order 
2". If a non-cyclic group G of order 2*" contains an odd num- 
ber of cyclic subgroups of order 2", at least one of these is in- 
variant under G. If this were the only cyclic subgroup of order 
2 in G there could not occur in G two cychc subgroups of order 
2'^^, /3>0, since one of these would be transformed into itself 
by one of its conjugates, and hence one of these and some 
operator in a conjugate subgroup would generate a group of 
order 2^+^+^ which would be conformal with the abelian group of 
t3^e {a-i-fi, 1). As this group would involve two cyclic groups 
of order 2", it has been proved that if G contains only one cyclic 
subgroup of order 2", it contains no more than one cyclic sub- 
group of order 2"+^, a-{-^<m, /3f 0. 

* Proceedings of the London Mathematical Society, vol. 2 (1904), p. 142. 



180 PRIME-POWER GROUPS [Ch. V 

Suppose that G contains one and only one cyclic subgroup 
of order 2** and that its largest cyclic subgroup H' is of order 
2". Since every subgroup of order 4 in the group of isomorph- 
isms of W involves the operator of order 2 which is com- 
mutative with just half of the operators of H^ it results that 
ai=m 1, otherwise G would involve a subgroup which would 
be conformal with the abelian group of type (ai, 1). From 
the known properties of the groups of order 2" which involve 
operators of order 2*""^ and the result just obtained, it follows 
that G must be one of the three groups mentioned above 
whenever it contains one and only one cycUc subgroup of order 
2. It remains to prove that G can involve only one such 
subgroup whenever it involves an odd number of cyclic sub- 
groups of this order. 

Suppose that G contains 2n+l cyclic subgroups of order 2**. 
At least one of them, Hi, is invariant under G. Let H2 repre- 
sent any other and let H'2 be a subgroup of H2 such that the 
group (^1, H'2) is of order 2"+^ Hence {Hi, H2') has just 
two cyclic subgroups of order 2" and at least 2**"^ invariant 
operators. These two cyclic subgroups have just 2""^ common 
operators and we proceed to prove that there is an even num- 
ber of cyclic subgroups of order 2 in G such that they all have 
2"~* operators in common. 

If there is in G another cyclic subgroup of order 2** which 
has exactly 2"^ operators in common with H\, it is trans- 
formed either into itself by (^1, H'2) or it is transformed into 
a power of two distinct conjugates such that all of these have 
2**"^ operators in common with H\. In the former case its 
generators are either invariant under all the operators of order 
2" in (^1, H'2) or they are transformed by at least some of 
these operators into themselves multiplied by the operators 
of order 2 in Hi, while the remaining operators of {Hi, H'2) 
transform each of these generators into themselves. In this 
case (^1, H'2) is therefore invariant under these subgroups, 
that is, {Hi, H'2) is invariant under all the subgroups of order 
2", contained in G, which have 2""^ operators in common with 
Hi and are transformed into themselves by (Hi, H'2). 



52l NUMBER OF NON-CYCLIC SUBGROUPS 131 

If Hz is such a subgroup it is clear that (//i, '2^ B3) is 
conformal with an abelian group and hence involves an even 
number of cyclic subgroups of order 2". We consider now all 
the other cyclic subgroups of order 2 contained in G, which 
have 2"~* operators in common with Hi and are transformed 
into themselves by (Hi, H'2, Ha)- If such a subgroup 7/4 
exists it is clear that (Hi, H'2, H3, H4) must again be conformal 
with an abelian group and hence it involves an even number 
of cyclic subgroups of order 2". By continuing this process 
it results that an even number of cyclic subgroups of order 2**, 
contained in G, have 2"~^ operators in common with Hi. 

All of these cyclic subgroups of order 2" form a set. If G 
contains a cyclic subgroup of order 2" which is not in this set, 
this subgroup must belong to another set which involves an 
even number of distinct cycb'c subgroups of order 2"; that is, 
these cyclic subgroups of order 2" are found in distinct sets 
such that no two have a subgroup in common and such that 
each set involves an even number of cyclic subgroups. This 
completes the proof of the theorem: if a group of order 2"* 
involves an odd number of cyclic subgroups of order 2, a>2, 
this number is unity. Combining this with the theorem proved 
above, it results that if we include the cyclic group there are 
just four groups of order 2*" such that each involves an odd number 
of cyclic subgroups of order 2", a>2.* In each of these groups 
there is only one such subgroup. 

The four groups mentioned in the last theorem contain one, 
2w-2_j_j Qj. 2-3-j-i cyclic subgroups of order 4. We proceed 
to prove that these are the only groups of order 2" which con- 
tain an odd number of cyclic subgroups of order 4. The method 
of proof is similar to the one employed above. 

Let G be any group of order 2" which contains an odd num- 
ber of cyclic subgroups of order 4. At least one of these sub- 
groups (Ki) is invariant under G, and at least half the oper- 
ators of G are conmiutative with a generator (s) of Ki. It 
will be proved that the subgroup Gi formed by these 2*"' 
operators must be cyclic. If Gi were not cyclic, A'l would be 

* Transactions of the American Mathematical Society, vol. 6 (1905), p. 58. 



132 PRIME-POWER GROUPS [Ch. V 

contained in an abclian subgroup of type (2, 1). This sub- 
group would contain two cyclic subgroups Ki and K2 of order 
4, having a common square. It will now be proved that G 
would then contain an even number of cyclic subgroups of order 
4 having s^ in common. 

If this number were odd, (Ki, K2) would transform at least 
one of the remaining ones (/V3) into itself. The commutator 
subgroup of {Ki, K2, Kz) is generated by s^ and hence this 
group contains an even number of cyclic subgroups of order 4 
and all of these subgroups contain s"^. Hence there would 
be another invariant cyclic subgroup K^ involving s^. The 
number of cyclic subgroups of order 4 in {Ki, K2, K3, K4) is 
again even, since the commutator subgroup is still generated 
by S' and all of these subgroups include s^. This follows almost 
directly from the fact that the product of an operator of order 
4 in (Ki, K2, K3, K4) into an operator of order 2 is of order 4 
when the two factors are commutative, and of order 2 when 
they are not commutative, while the converse is true if the 
second operator is of order 4. 

As this process could be continued indefinitely if there were 
an odd number of cyclic subgroups of order 4 which contained 
s^, it results that the number of these subgroups is even. If 
there were an odd number of cyclic subgroups of order 4 in G, 
which did not contain s^, one of these and s^ would again 
generate the abelian group of type (2, 1), and the number of 
those involving the same subgroup of order 2 would again be 
even. As similar remarks would apply to all the other possible 
cyclic subgroups of order 4 we have proved that Gi is cyclic 
whenever G contains an odd number of cyclic subgroups of order 4. 
This proves the statement in the first paragraph of this section, 
since the other non-cyclic group of order 2*" which contains 
a cyclic subgroup of order 2*""^ contains an even number of 
cyclic subgroups of order 4. 

Combining the results of this section we conclude that every 
group of order 2", w>3, which contains an odd number of 
cyclic subgroups of order 4, contains just one cyclic subgroup 
of order 2", where a can have any value from 3 to m 1; and 



52] NUMBER OF NON-CYCLIC SUBGROUPS 133 

every group which contains only one cyclic subgroup of order 
2" contains an odd number of cyclic subgroups of order 4. For 
each value of a and m there are three such groups; hence there 
is an infinite system of groups of order 2*" which contain an odd 
number of cyclic subgroups of composite order. When w = 3 
there are only two groups having this property, viz., the 
quaternion group and the group of movements of the square. 
If all the non-cyclic groups of order p^{m>Z, p an arbitrary 
prime) were determined, there would be just three among them 
in which the number of cyclic subgroups of composite order 
would not always be a multiple of p. In these three special 
cases the number of cycHc subgroups of every composite order 
is not divisible by p. 

In the exceptional groups noted above the number of the 
subgroups of order 2 is =1 (mod 4). That this number is 
=3 (mod 4) in every other non-cyclic group of order 2"* is a 
direct consequence of the fact that the number of cych'c sub- 
groups of order 4 in all these groups is even. From this fact 
it results that the number of operators whose order exceeds 
2 is divisible by 4, since every cyclic subgroup of order 2' con- 
tains 2'~^ operators which are not found in any other subgroup 
whose order is = 2'. Hence the given system of groups is 
composed of all the groups of order p*^ in which the number 
of subgroups of order p is not =l-\-p (mod p^). That is, the 
groups of order /?*" in which the number of cyclic subgro\ips of 
composite order is not divisible by p coincide with those in which 
the number of subgroups of order p is not of the form l-\-p-\-kp^. 

EXERCISES 

1. The number of subgroups of order p in any non-cyclic group of order 
/", p>2, is of the form l+p+kp\ 

Suggestion: Consider the forms of the number of the operators of orders 
p\p', etc. 

2. The number of cyclic groups of order p^,p>2 and /3> 1 , is of the form 
kp whenever the Sylow subgroups of order />"* are non-cyclic. 

3. If a group G of order p"* contains exactly p cyclic subgroups of order 
p", these subgroups generate a characteristic subgroup of order p""*"^ 
under G, and this subgroup is either abelian and of type (a, I) or it is the 
non-abelian group which is conformal with this abelian group. 



134 PKlME-ruWEli GROUPS [Ch. V 

4. There are just three groups of order 3* each of which contains 
only three cyclic subgroups of order 9. 

53. Some Properties of the Group of Isomorphisms of a Group 
of Order p^. If Ga is any invariant subgroup of G, then Ga 
contains at least one invariant subgroup of G whose order is 
an arbitrary divisor of the order of Ga- Let 

Go, Gi, G2, . . . , Gm 

be a series of invariant subgroups of G whose orders are respect- 
ively 1, P, p"^, ' ' , P^, so that Gm=G; and suppose that each 
of these subgroups except the last is included in the one which 
follows it. Let / be any operator whose order is a power of p 
in the group of isomorphisms of G. Such an operator is said 
to effect a p-isomorphism of G. Since / and G generate a group 
whose order is a power of p, we may suppose that / transforms 
all of the operators of each one of the given series of subgroups 
into themselves multiphed by those of the preceding subgroup. 
That is, the commutator of t and any operator of G^ is in G^-i, 
i3 = l, 2, . . . ,w. 

We proceed to prove that the order of / cannot exceed p*^~^. 
In fact, if Ifi is any operator of Gp in the above series, it results 
from the given conditions that 

Hence /***" must be commutative with every operator of G, 
and, as t is an operator in the group of isomorphisms of G, it 
results from this that the order of / is a divisor of p^~^. In 
other words, the p^'^ power oj every operator whose order is a 
power of p in the group of isomorphisms of a group oJ order p^ 
is the identity. When G is the cycUc group of order p^, p>2, 
it is known that the group of isomorphisms of G involves oper- 
ators of order p^~^, viz., those operators which give rise to 
commutators of order pr^~^. 

We proceed to prove that the group of isomorphisms of a 
non-cyclic group of order />"*, p>2 and w>3, cannot involve 
any operator of order p^'"^. If G is such a group it follows from 
the preceding section that we may assume that G2, in the above 



S 631 THEIR GROUP OF ISOMORPHISMS 135 

series of invariant subgroups, is non-cyclic. If / is an operator 
in the group of isomorphisms of G and if the order of / is a power 
of p, it results that /" is commutative with every operator of 
G2 and that it transforms the operators of G into themselves 
multiplied by operators in Gm-2. Similarly /"* is commutative 
with every operator in G3 and transforms every operator in Gm 
into itself multiplied by an operator in Gm-3- In general, /"* 
transforms every operator in G into itself multiplied by operators 
in Gn,-a-i- As t" transforms the operators of G into itself 
multiplied by operators in G2 and as these operators are com- 
mutative under this power of t, it results that 

1-2 
t'' =1. 

That is, the order of every operator in a Sylow subgroup of order 
p^ of the group of isomorphisms of a group of order />" is a divisor 
of />*"* whenever p> 2 and m> 3. 

The last theorem evidently also applies to all groups of order 
2" which involve a non-cyclic invariant subgroup of order 4, 
and it is well known to be true as regards the cyclic group of 
order 2*". It is, however, not true as regards the dihedral or 
the dicyclic group of order 2", since there is evidently an opera- 
tor of order 2"*~^ in the group of isomorphisms of each of these 
groups; viz., the operator which is commutative with each 
operator of the cyclic subgroup of order 2"*"^ but transforms 
the non-invariant operators of orders 2 and 4 respectively into 
themselves multipUed by an operator of order 2**~^ These 
two infinite systems of groups therefore are instances of groups 
whose groups of isomorphisms contain operators of the largest 
possible order in accord with the theorem of the second para- 
graph of the present section. 

As the remaining group of order 2" which does not involve 
an invariant non-cyclic subgroup of order 4 involves 2"*"^ non- 
invariant cyclic subgroups of order 4, its group of isomorphisms 
may be represented as an intransitive substitution group on 
2""^ letters. From this it follows directly that its group of 
isomorphisms cannot involve any operator of order 2""*, 
m>3. Hence we have proved the theorem: 



136 PRIME-POWER GROUPS (\i. V 

The only groups of order />"*, m> 3, wJwse groups oj isomorph- 
isms involve operators of order p"*~^ are the cyclic group when 
P>2y and the dihedral and the dicyclic groups when p = 2. 

Every operator whose order is a power of /> in the group of 
isomorphisms of G generates with G a group whose order is a 
pKJwer of p. As G is an invariant subgroup of this group it 
results that this group involves a series of invariant subgroups 

Go, Gi, G2, . . . , Gm 

which are such that / transforms the operators of each of these 
subgroups into themselves multiplied by those of a preceding 
subgroup. Moreover, whenever / has this property its order is 
a power of p. Hence the existence of such a series of subgroups 
is a necessary and sufficient condition that an operator / in 
the group of isomorphisms of G has for its order a power of p. 
In other words, 

A necessary and sufficient condition that an operator t in the 
group of isomorphisms of a group G of order p"* has for its order 
a power of p is that t transform every operator in tJie series of sub- 
groups 

Go, Gi, G2, ' ' , Gm = G 

of orders I, p, p^, . . _. , />"* respectively, into itself multiplied by 
an operator in the preceding subgroup. 

Hence t^ transforms every operator of Gp into itself mul- 
tiplied by an operator inG^-a-i, where jS a 1 is to be replaced 
by whenever i3a+l. 

From this theorem it is easy to deduce the following useful 
corollary: The order of the commutator of two operators, neither 
of which is the identity, of a group G of order p^ is less than the 
order of the smallest invariant subgroup of G in which either of 
these two operators occurs. In particular, if one of these two 
operators is of order p and generates an invariant subgroup, it 
must be commutative with every operator of G. This special 
case may also be regarded as a special case of the theorem that 
every invariant subgroup, besides the identity, in a group of 
order p^ involves invariant operators of order p. 



54J SYLOW SUBGROUPS OF ISOMORPHISMS 137 

The group of isomorphisms of the group of order p is the 
cyclic group of order /> 1, and hence it does not involve any 
operator of order p. It is easy to prove that the group of iso- 
morphisms of every group G of order p^, m>\, involves oper- 
ators of order p. In fact, when G is non-abelian its group of 
cogredient isomorphisms involves such operators and when 
G is abelian we may select any subgroup of order p^'^, and 
multiply the operators of the corresponding quotient group by 
those of an invariant subgroup of order p which is contained in 
the given subgroup of order p^~^. It is evident that the latter 
isomorphisms are also possible even when the group of order 
p^ is non-abelian. Hence the group of order p is the only 
group of order />"* whose group of isomorphisms does not in- 
volve operators of order p. 

EXERCISES 

1. If G is the abelian group of order />*" and of type (1, 1, . . . ), its 
group of isomorphisms cannot involve any operator whose order is a power 
of p and exceeds p^^^ when m is even or ^<'"+i)/2 when m is odd. 

Suggestion: When a power of / transforms G into itself multiplied 
only by operators which are commutative with t, the ^th power of this 
power of t will be the identity. 

2. The group of isomorphisms of the dihedral group of order 2, >2, 
is the holomorph of the cyclic group of order n. 

Suggestion. The group of isomorphisms of this dihedral group may 
be represented as a transitive substitution group of degree , and it in- 
volves an invariant cyclic subgroup of order n composed of all its opera- 
tors which are commutative with every operator of the cyclic subgroup 
of order n. 

3. Find the group of isomorphisms of the dicyclic group of order 2", 
n>3. 

4. If a group of order 2"* contains exactly two cyclic subgroups of order 
2^, but no cyclic subgroup of any higher order, then m/3-f2. 

64. Maximal Order of a Sylow Subgroup in the Group of 
Isomorphism of a Group of Order p"*. We proceed to deter- 
mine the maximal order of a Sylow subgroup of order />* con- 
tained in the group of isomorphisms of a group of order p^. 
To do this we assume again that a set of invariant subgroups 



138 PRLME-roWKR GROUPS (Ch. V 

of orders I, p, p^, - - , p^ respectively of such a group G is 
represented by the symbols 

Go, Gi, G2, I Gmi 

and that these invariant subgroups have the property that each 
one except the last is included in the one which follows it. The 
operators of G which are not also in Gm-i may be transformed 
into themselves multiplied by at most p^~^ different operators. 
In general, the operators of Ga which are not in Ga-i are 
transformed into themselves multiplied by at most p"'^ opera- 
tors. Hence the order of a Sylow subgroup of order p^ in 
the group of isomorphisms of G cannot exceed 

p.p2 . . ^ . , pm-l _pm(.m-l)/2^ 

It is evident that when G is abelian and of type (1, 1, . . . ), 
its group of isomorphisms involves Sylow subgroups of order 
/)", where n has the maximal value given above. This is also 
the case when G is abelian and of type (2, 1, 1, ... ). It 
is not difl&cult to prove that these are the only abelian groups 
whose groups of isomorphisms contain Sylow subgroups of the 
given maximal order.* 

56. Construction of all the Possible Groups of Order p"". 
In view of the simple properties of the group of order p"* various 
efforts have been made to determine all these groups for given 
values of m. When m = l there is evidently only one such 
group, and when m = 2 there are two possible groups. Each 
of these groups is abelian. Moreover, it has been observed 
that for any value of m the number of abelian groups is equal 
to the number of different partitions of m as regards addition. 
Hence the only difficulty in the determination of all these groups 
for small values of m relates to the possible non-abelian groups. 
In case m<5 there is very little difficulty here, but for larger 
values of m this difficulty increases very rapidly with m. 

As an abstract group is simply isomorphic with one and 
only one regular substitution group, we can determine all the 

* Cf. G. A. Miller, Transactions of the American Malhematical Society, vol. 
12 (1911) p. 396. 



65) CONSTRUCTION OF GROUPS OF ORDER p" 139 

possible groups of order />" by determining all the regular 
substitution groups of this order. We proceed to indicate a 
method for constructing all the possible regular groups of 
order p^ on condition that all those of order p^~^ are known. 
Since every regular group of order p^ contains an invariant 
intransitive subgroup of order />"*~^ which is formed by a simple 
isomorphism between p regular groups of this order, we may 
first determine all the possible regular groups of order />" which 
involve a given group B. of order p^~'^. 

Let the p transitive constituents of H he Hi, H2, . . . , B. 
A group G of order />*" which contains H is generated by H and 
some substitution h which permutes the p systems of intran- 
sitivity of H cyclically, has its />th power in H, and transforms 
E into itself. Let / be a substitution of order p which permutes 
the systems of E transitively and is commutative with every 
substitution of E. As it may be assumed that /i and t permute 
these systems in the same way, we conclude that ht-^ is a sub- 
stitution 5 which does not permute any of the p transitive con- 
stituents of E. That is, tx =st, where it may be assumed that t 
transforms the corresponding letters of each of the p systems 
among themselves. As / can readily be obtained from the p 
constituents belonging to Ei, E2, . . . , Ep oi any substi- 
tution in E it remains only to determine s = siS2 . . . Sg, where 
Si, S2, ' ' , Sp are the constituents of s belonging to Hi, E2, 
. . . , Ep respectively. As Si, S2, . . . , Sp transform each 
of the constituents Ei, E2, . . . , Ep into itself they must be 
found in the holomorphs of Ei, E21 . , Ep, respectively. 

We proceed to prove that 52, Sz, . . . , Sp may be assumed 
to be corresponding substitutions in the groups of isomorphisms 
of E21 Ez, . . . , Ep, respectively, while ^1 is one of the />"*~^ 
substitutions obtained by multiplying the substitution in the 
group of isomorphisms of Ei which corresponds to ^2, 53, . . . , 
Sp by the f^~^ substitutions which are commutative with 
El and involve only letters of Ei. This theorem follows 
almost directly from the fact that the conjoint of a regular 
group has the same group of isomorphisms as the regular group 
itself, since both are invariant subgroups of the holomorph of 



140 PRIME-POWER GROUPS [Ch. V 

this regular group, and the group of isomorphisms of a regular 
group is the subgroup composed of all the substitutions which 
omit a given letter in its holomorph. Hence we may reduce 
each of the substitutions which transform // in a certain way 
to one of the given form by transforming by a substitution in 
the direct product of the conjoints of H\, H2, . . , Up. 

From the preceding paragraph it results that the number 
of groups of order p"* which contain a given group 77 of order 
pm-i cannot exceed the order of the product of H and the 
order of its group of isomorphisms. Hence the number of the 
groups of order p'^ is always finite when p is finite. We shall 
be able, however, to reduce this apparently possible number 
very greatly. In the first place it should be observed that if 
5/ is so selected that Si'soSs ... 5p is commutative with /, 
while si involves only letters in Hi, then si=si'^si where 
5i" is commutative- with si'; otherwise (51^2 . . . Spty would 
not be in H. This restricts the number of choices of si" to the 
number of operators in H which are both in its conjoint and 
also inviariant under si. Another important restriction is that 
we evidently need to consider only one of the first p l powers 
of 5/5253 . . . Sp, since / may be transformed into any power 
without affecting 5i52 ... 5p by permuting the constituents 
H2, Hs, . . . , Hp cycUcally. When H is abelian, all the groups 
obtained by replacing 5/' by any of its powers prime to its order 
are conjugate, since the operators in the group of isomorphisms 
of an abelian group which transform each operator of this group 
into the same power are invariant under this group of iso- 
morphisms. 

To illustrate the method outlined above we proceed to 
determine the possible non-abelian groups of order p^, p>2. 
It has been observed that each of these groups involves a non- 
cyclic group of order p^, and hence we may use this for H. As 
the group of isomorphisms of // involves only one set of con- 
jugate subgroups of order />, the two substitutions 5/5253 . . . Sp 
and t may be supposed to be the same for all these groups. 
Moreover, 5/' is restricted to a single subgroup, and hence it 
is necessary to consider only two cases, viz., the case when 



551 CONSTRUCTION OF GROUPS OF ORDER p* 141 

Si" is the identity and the case when its order is p. That is, 
there cannot be more than two non-ahelian groups of order 
p^, p>2, in accord with the general theory which precedes 
this paragraph. From the present paragraph it results that 
one of these groups involves operators of order p^ while the 
other does not have this property. Hence there are two and 
only two non-abelian groups of order p^, p>2. It was proved 
in 33 that there are also only two such groups when p = 2. 

Among the groups of order p" which involve H the one gen- 
erated by H and / is of especial interest in view of its simple 
structure. In fact, it is abstractly the direct product of H and 
the group of order />. If we transform this group by 

ri^-^nu^ . . . r/-^ 

where ri is any invariant substitution of Hi while r^, a = 3, . . . , 
p, is the transform of ri with respect to t", there results a group 
generated by H and ri~^t. From this and the preceding theory 
it results that we need to use only two values for si" when 
n is cycUc and the entire group is abelian, as results also directly 
from the theory of the abelian groups. This general method 
can easily be extended so as to reduce the amount of labor 
necessary to determine all the possible groups of order f^ 
which involve a given subgroup of order p""'^ *; but the pre- 
ceding developments may suffice to point the way towards a 
penetration into this difficult subject. From the given illus- 
tration it results that we do not always need to use all the pos- 
sible groups of order />"*~^ for // in order to determine all the 
groups of order p^. In fact, when w = 4 we need to consider 
only the abelian groups of order p^ for H, since every group of 
order p* contains an abelian subgroup of order p^. 

EXERCISES 

1. Determine the fourteen possible groups of order 16. 

2. Determine the non-abelian groups of order />*, />>2, which involve 
no operator of order />-. Show that Si"= 1 in these cases and that there 
are two such groups when />>3, one having a commutator subgroup of 

* Aitkirican Journal of Mathematics, vol. 24 (1902), p. 395. 



142 PRIME-POWER GROUPS [Cn. V 

order p and the other having such a subgroup of order p^. When ^=3 
there is only one such group. 

3. There are four distinct non-abelian groups of order />*, />>3, which 
involve an abelian group of order />*, but do not contain any operator of 
order ^*. When /=3 there are only two such groups. 

4. The number m can be so chosen that the number of the distinct 
groups of order />'", />>2, which do not involve any operator of order p^ 
is greater than any given number. 

5. Every intransitive Sylow subgroup of a symmetric group is the 
direct product of its transitive constituents, and each of these transitive 
constituents has a central of prime order. 

6. If the degree of a symmetric group is =i&i^**+;fe2/'*~^+ . . 
+*a+i; *i' h, . . . , ka+\ being positive integers less than p, then the 
central of its Sylow subgroup of order p^ is of order />*+*+ +*. 



CHAPTER VI 

GROUPS HAVING SIMPLE ABSTRACT DEFINITIONS 

56. Groups Generated by Two Operators Having a Common 
Square. If si, S2 represent two operators of order 2 they evi- 
dently satisfy the equation Si^ = S2^, and the cyclic group (S1S2) 
generated by S1S2 is invariant under ^i and 52. In fact, S1S2 
is transformed into its inverse by each of the two operators 
Si, 52, and hence (^i, 52) is the dihedral group whose order is 
the double of the order of 51^2, as has been observed in 26. 
Hence the equations 

51^=52^ = 1, (5i52)=l 

serve as a complete definition of the dihedral group of order 
2, if we assume that the order of 51^2 is exactly n. Through- 
out the present chapter it will be assumed, unless the contrary 
is stated, that the condition 51"= 1 implies that the order of 
Si is exactly n. This fact is sometimes expressed by saying 
that si fulfils the condition si* = l, while the statement 5i 
satisfies the condition 5i" = 1 may imply merely that the order 
of Si is a divisor of n* We shall employ the terms fulfil and 
satisfy with these meanings throughout the present chapter, 
so that the equation 51" = 1 implies that si fulfils this condition 
unless the contrary is stated. 

Ji Si, S2 are any two operators which satisfy the equation 

Sl^=S2^, 

they generate a group G under which Si^ is invariant. That 
is, the cyclic group generated by ^i^ is composed of operators 
which are invariant under G, since Si^ is commutative with each 

Quarterly Journal of Maiketnatics, vol. 41 (1909-10), p. 169. 
143 



144 GROUPS DEFINED ABSTRACTLY [Ch. VI 

of the op>erators si and S2. From the fact that 51^ = 52^, it 
results that 

Each of the operators siS2~^, ^2^1"^ is therefore transformed 
into the other by each of the two operators si, S2. That is, 
the cyclic group generated by either of these operators is 
invariant under G, and each of its operators is transformed into 
its inverse by si as well as by S2- The abelian group generated 
by the two operators si^, SiS2~^ must therefore be invariant 
under G, and it involves either all, or just half of the operators 

of a 

When each of the two operators si, S2 is of odd order, G is 
cyclic, since each of these operators is equal to the other, and 
G is generated by ^i^ in this case. When one of these operators 
is of odd order while the other is of even order, G is generated 
by the operator of even order and 51^2 ~^ is of order 2. The 
only case which requires further consideration is therefore 
the one in which the common order of Si, S2 is an even number 
2. A necessary and sufficient condition that G be abelian 
is that the order of SiS2~^ divide 2. If siS2~^ = l, Si=S2 and 
G is the cyclic group generated by Si. If the order of 5i52"^ 
is 2, G is either the cyclic group of order 2 or the abelian group 
whose invariants are 2 and 2n. It remains only to consider 
the cases when G is non-abelian; that is, when the order of 
SiS2~^ exceeds 2, and hence Si, S2 have the same even order 2n. 

It is evident that the common order of ^i, ^2 is not limited 
by the relation 51^ = 52^. That is, for an arbitrary value of n 
we can find two operators Si, S2 of order 2w such that they satisfy 
the equation Si^ = S2^. In fact, the value of n is not limited if 
we impose the additional condition that the order of 5i52~^ 
shall be an arbitrary number m, since two generators of order 
2 of the dihedral group of order 2m may be multiplied by an 
operator of order 2 which is commutative with both of these 
generators such that the products satisfy both of these con- 
ditions. In other words, for any arbitrary number pair m, 
n, we can find two operators ^i, S2 of order 2n such that they 



56] TWO GENERATORS HAVING A COMMON SQUARE 145 

have a common square and that the 'order of SiS2~^ is m. 
The order of the group generated by these two operators is 
always a divisor of 2mn and a multiple of mn, since ^i^ and 
siS2~^, are of orders n and m respectively and the cyclic group 
generated by these operators cannot have more than two 
common operators. 

It results from the preceding paragraph that G is completely 
defined by the three conditions 

whenever either w or is odd. In fact, when si, S2 fulfil these 
conditions and either w or is an odd number, G is the group 
of order 2mn obtained by establishing an {m, n) isomorphism 
between the dihedral group of order 2m and the cyclic group 
of order 2w. If m and n are both even, G may again be con- 
structed by establishing such an {m, n) isomorphism when 
the order of G is 2mn. WTien this order is only mn, G may 
be constructed by establishing a simple isomorphism between 
n cyclic groups of degree and order w, and then extending this 
group by means of an operator of order 2n which transforms 
into its inverse each operator of this cyclic subgroup, permutes 
its systems of intransitivity, and has its wth power in this sub- 
group. That is, the two operators 5i, ^2 may he so selected as to 
fulM the three conditions 

Sl^=S2^ (5l52-^)* = l, 51^ = 1 

and to generate either of two groups when m and n are both given 
even numbers. When at least one of them is a given odd number, 
the group generated by Si, 52 is completely determined by the three 
given relations. 

To illustrate this theorem we begin with the case when 
m = 4 and = 2. The group of order mn in this case is clearly 
the quaternion group, while the group of order 2mn may be 
constructed by establishing a (4, 2) isomorphism between the 
octic group and the cyclic group of order 4 so as to obtain the 
group of order IG involving 12 operators of order 4 which have 
only two distinct squares. This group has the quaternion 



146 GROUPS DEFINED ABSTRACTLY [Ch. VI 

group for a quotient group but not for a subgroup. If a set 
of generating operators satisfy certain conditions, the largest 
group which they may generate has all the other groups which 
they may generate for quotient groups. This follows directly 
from the theory of the quotient group. When m = n = 2, G 
is clearly abelian and is of order 4 or 8. In the former case it 
is cyclic and in the latter it is of type (2, 1). When m = 3 
and w = 2, G is the dicyclic group of order 12. 

When n = l, the category of groups under consideration 
clearly coincides with the dihedral groups. When n = 2 and m 
is odd it coincides with all the dicyclic groups whose orders 
are not divisible by 8, and when w = 2 and m is even, the groups 
of order inn coincide with the totality of the dicyclic groups^ 
while those of order 2mn may be obtained by establishing an 
(w, 2) isomorphism between either the dihedral group of order 
2m or the dicyclic group of order 2m and the cyclic group of 
order 4. Hence the dihedral groups and the dicyclic groups 
may both be regarded as special cases of groups generated 
by two operators having a common square. Since ^1^2 = 
SiS2~^S2^, it results that the order of the product of the two 
operators ^1 and 52 is either the least common multiple between 
m and , or it is exactly half this least common multiple. That 
is, the order of the product of two operators is not restricted 
by the fact that they have a common square, and the order 
of the group which they generate is always a divisor of the 
double of the square of the order of this product. 

If we let /i=5i and t2 = siS2~^, it results that the group 
(/i, 12) is identical with (51, 52). That is, every group that can 
be generated by two operators having a common square can also 
be generated by two operators such that the one transforms the 
other into its inverse, and vice versa. Hence we have two abstract 
definitions for this category of groups. The latter definition 
is more convenient than the former for the purpose of obtain- 
ing directly the abstract properties of these groups, but in the 
abstract theory these groups frequently present themselves 
under the former definition, and hence it is very important 
to know that the two given definitions apply to the same cate- 



57] GROUPS OF THE REGULAR POLYHEDRONS 147 

gory of groups. It results directly from the given equation 
that S2 = t2'Hi. That is, if trH2ti=t2-^ then ti^ = (t2-Hiy. 
In the special case of the dihedral groups the two equivalent 
definitions reduce to 

EXERCISES 

1. If Si, Si are two operators, neither of which is the identity, which 
satisfy both of the conditions: 

they must also satisfy the conditions: 

Si~^S2Si = S2^, S2~^SiS2=Si*; 

and they generate either the quaternion group or the four-group. 

2. Find the number of operators of order 6 in the group of order g 
which is generated by Si, S2 subject to the following conditions: 5i= 1, 
Si~^SiSi=S2~^. Prove that the central of this group is either of order 
3 or of order 6, and that g may be an arbitrary multiple of 6. 

67. Groups of the Regular Polyhedrons.* The regular 
tetrahedron evidently admits two movements of order 3 whose 
product is of order 2. If we represent these movements by Si, 
S2 respectively, these operators must fulfil the conditions 

5i3=52^ = (5i52p = l, or 51^=52^=1, SiS2 = S2^Si^. 

We proceed to prove that these conditions define a group of 
order 12; whence, as the group of movements of the regular 
tetrahedron is evidently of order 12, the group defined by the 
given two equivalent sets of conditions must be the group of the 
regular tetrahedron. To prove that (si, S2) is of order 12 
we may proceed as follows: The three conjugate operators 
of order 2 

SlS2, S2S1, Si^S2Sl^ 

are commutative, since 

525 1%2 = Si^S2^ ' S2^S\^ = S\^S25\^. 

* These groups were first studied by means of abstract definitions by W. R, 
Hamilton, cf. Bibliotheca Mathemalka, vol. 11 (1910-11), p. 314. 



148 GROUPS DEFINED ABSTRACTLY (Ch. VI 

As two operators of order 2 whose product is of order 2 gener- 
ate the four-group, it results that {si, S2) involves the four- 
group represented by the operators 

1, S1S2, S2S1, Si^S2Si^. 

As this subgroup is invariant under si as well as under 51^2, 
it must be invariant under 52, and therefore also under (^i, 52). 
Hence the group generated by Si and this subgroup of order 
4 is of order 12. Since this group involves ^1 and 51J2, it must 
also involve 52; that is, it must be {si, 52). This proves the 
theorem : 

// the order of the product of two operators of order 3 is of 
order 2 they generate the tetrahedral group. 

If we replace Ji by /i and s\S2 by t2, it results that /i, ti-'^ti 
are two operators of order 3 whose product is of order 2; hence 
{t\, tr^ti) is the tetrahedral group. Since (/i, ti-^t2) = {ti, /2) 
it results also that if the order of the product of two operators 
of orders 2 and 3 respectively is 3, then these operators must 
generate the tetrahedral group. That is, Si, S2 generate the 
tetrahedral group if they fulfil either one of the following two 
sets of conditions: 

Si^ = S2^ = (SiS2y = 1, Si^=S2^ = (SiS2y = 1. 

These two sets of equations furnish two very useful definitions 
of this important group. The group could also be defined by 
the facts that its order is 12 and that it does not involve a sub- 
group of order 6, as well as by the facts that it is of order 12 and 
contains four subgroups of order 3. 

The cube is clearly transformed into itself by 24 movements 
of rigid space, and the order of each of these movements is 
equal to one of the four numbers 1, 2, 3, and 4. It is not diffi- 
cult to find, among these 24 movements, two of orders 3 and 4 
respectively whose product is of order 2. If we represent these 
tv;o movements by Si, 52, they must therefore satisfy the equa- 
tions 

5l3=52* = (5i52)2 = l. 



67) GROUPS OF THE REGULAR POLYHEDRONS 149 

We proceed to prove that these conditional equations define 
a group of order 24, and hence they must define the group of 
the cube. To prove that they define a group of order 24 we 
may proceed as follows: The group (51, S2^) is the tetrahedral 
group, as results directly from the fact that 

{SiS2^y = 5i52 S2S1S2 S2S1S2 So = S1S2 'Si^'5i^-S2 = l. 

This tetrahedral group is invariant under 52, since 

S2~^SiS2 = 52%l52 = 52^ S2S1S2 = So^Si^. 

As (5 1, S2^) is invariant under S2 and involves S2^, it follows 
that (^1, S2) is of order 24. That is, 

Two operators of orders 3 and 4 respectively whose product 
is of order 2 generate the group of the cube. 

From the given equations it results directly that this theorem 
may also be stated in either of the following two ways: Two 
operators of orders 2 and 3 respectively whose product is of 
order 4 generate the group of the cube, or two operators of 
orders 2 and 4 respectively whose product is of order 3 generate 
the group of the cube. The group of the cube may therefore 
be defined as the group generated by 5 1, 52 when these operators 
fulfil any one of the following four sets of equations: 

Sl^=S2^ = {SiS2Y = \; 51^ =52'* = !, SxS2=S2V\ 
5i2 =523 = {s^SiY = 1 ; Sx^ =52^ = {siSiY = 1. 

The group of the cube is also known as the octahedron group, 
in view of the fact that the group of movements of the regular 
octahedron coincides with that of the cube. This group presents 
itself in many investigations and has been defined abstractly 
in a number of other ways. Among these are the following: 
Two operators of order 4 generate the octahedron group pro- 
vided their product is of order 3 and each of them is trans- 
formed into its inverse by the square of the other. The octa- 
hedron group is the smallest group that can be generated by 
two operators whose orders exceed 2 and which are such that 
each is transformed into its inverse by the square of the other.* 

* Annds of Mathematics, vol. 21 (l'J07), p. 50. 



150 GROUPS DEFINED ABSTRACTLY ICh. VI 

The octahedron group is completely defined by the fact that it 
can be generated by three cyclic non-invariant subgroups of 
order 4 which do not involve a common subgroup of order 2 
nor generate other operators of order 4.* The octahedron 
group may be defined as the group which contains exactly 9 
operators of order 2 such that the order of the product of any 
two does not exceed 4, while there are at least two such opera- 
tors whose product is of order 4.t 

58. Group of the Regular Icosahedron. Both the regular 
icosahedron and the regular duodecahedron admit two move- 
ments of orders 2 and 5 respectively whose product is of order 
3. If we let 5i, 52 represent these movements it results that 
these solids are transformed into themselves by operators 
which fulfil the following equations: 

It is not diflScult to prove that these equations define the simple 
group of order 60 4 As the regular icosahedron and the regular 
duodecahedron admit exactly 60 movements, it will result from 
this proof that their group of movements must be the simple 
group of order 60. Hence this group is frequently called the 
icosahedron group. To prove that (^i, S2) is of order 60 if no 
restrictions are placed on these operators except those implied 
in the given equations, we may proceed as follows: 

The operators siS2^siS2^Si, siS2^siS2^si are of order 2, since 
they coincide with their inverses. They transform S2 into its 
inverse, since the product of S2 and either one of these opera- 
tors is of order 2, as may be seen directly if we employ the 
equation 5i525i=52^5iS2*, resulting from (si52)^ = l. From this 
equation it results also that 

{SiS2^)^ = {siS2^)^ = l. 

In fact, 

{SlS2Si)^=S2^{SlS2^)^S2=l, sinCC {815281)" = SlS2''Sl. 

* Mathematische Annalen, vol. 64 (1907), p. 344. 

t American Journal of Malhcmalks, vol. 29 (1907), p. 8. 

t Hamilton, Philosophical Magazine, vol. 12 (1856), p. 44fi. 



58] GROUP OF THE REGULAR IC08AHEDR0N 151 

As SiS-2^ is the transform of the inverse of siSi^, the given state- 
ment is proved. The most general product that can be formed 
with the operators Si, S2 is of the form 

If the number of factors in this product exceeds six it can evi- 
dently be reduced by means of one or more of the following 
equations: 

SlS2Sl=S2'^SiS2'^, SiS2'^Si=S2SlS2, (^1^2^)^= 1, 
SiS2^Si = S2*SiS2^SiS2'^, SiS2^Si = S2SlS2^SiS2. 

From these equations and from the fact that 

SiS2^SiS2^SiS2' = S2~SiS2^SiS2^Si 

it results that all the distinct operators of (51, S2) can be written 
in one of the following forms: 

52", S2'^SiS2'', S2'^SiS2^SiS2*', S2'"5i52%lS2^5i(w, = 1, 2, . . . , 5). 

Hence (51, S2) is of order 60 if we assume that si, S2 fulfil the 
equations given above. 

The three sets of equations 

Sl^=S2^={siS2y = l, Si^=S2^ = {siS2)^=l, Si^ =$2^ = {5152^ = 1 

are evidently equivalent and hence each of them defines the 
icosahedral group. In fact, if we should assume that the opera- 
tors 5i, 52 merely satisfy any one of these sets of equations, they 
would generate the icosahedron group unless both of them 
were the identity. Hence we have the theorem: 

// the three numbers 2, 3, 5 are the orders of two operators and 
of their product, these operators must generate the icosahedron 
group irrespective of which of these three numbers is the order 
of the product. 

EXERCISES 

1. If two operators merely satisfy the equations defining the tetra- 
hedral group they may generate the cyclic group of order 3, and if they 
merely satisfy the equations defining the octahedral group they may 
generate the symmetric group of order 6 or the group of order 2. 

2. If a group of order 12 contains no invariant operator of order 2 it 
must be tetrahedral. 



152 GROUPS DEFINED ABSTRACTLY (Ch. VI 

3. If two operators of or^er 3 have a product of order 2n whose square 
is invariant under both of these operators, they generate a group of order 
12/1 whose group of cogredient isomorphisms is the tetrahedral group. 

69. Generalizations of the Group of the Regular Tetra- 
hedron.* An immediate generalization of the tetrahedral 
group is given by the equations 

It results directly that 5i^ = / is invariant under (5i, 52) and that 
the group of cogredient isomorphisms of (51, S2) is the tetra- 
hedral group. From the fact that S1S2 and 52^1 are two opera- 
tors of order 2 it results that siS2^si is transformed into its 
inverse by each of the operators S1S2 and 52^1 . Moreover, 

{SiS2^Siy = SiS2^Si^S2^Si^Si~^ = Si.S2~^Si-^S2~^Si-^ .Si-^Si^^ = t* 

is both invariant and also transformed into its inverse under 
(51, S2). Hence the order of / is a divisor of 8. If this order is 
8, the order of G is 96. That there is such a group of order 9G 
may readily be seen by means of the two substitutions 

5i =ac'ce'eg'ga''hm'nfoj'hk'ld'mp'fi'jh'kn'do'ph'U', 

S2 = am' je'on'gk'pc'ml'ei'na'kj'co'lg'ip' hd'dJ'Jh'W. 

The existence of this group of order 96 and some of its prop- 
erties may also be established abstractly as follows: Let 
^'1, ^'2 be two operators of order 3 whose product is of order 4, 
and suppose that they have been so chosen that (51', 52') is the 
group of order 24 which does not involve any subgroup of 
order 12, i.e., the non-twelve f ^"24. If we extend this group 
by means of an operator t of order 8 which is commutative with 
each of its operators and generates its operators of order 2, 
it is clear that s\t, S2t may be used for S\, S2 respectively. 
That such an operator of order 8 actually exists can readily 

Some of these developments could have been presented more briefly by 
employing the theorem that if the square of an element of a commutator is com- 
mutative with the commutator it transforms this commutator into its inverse. 
The presentation adopted seems a little more elementary'. 
t American Journal 0} Mathematics, vol. 32 (1910), p. 65. 



5Ul GENERALIZATION OP TETRAHEDRAL GROUP 153 

be seen if the non-'twelve G24 is written in the regular form in 
four distinct sets of letters, and a simple isomorphism is estab- 
lished between these groups. The operator of older 4 which 
merely permutes cyclically the corresponding letters in 
these four constituent groups, multiplied by the operator of 
order 2 in one of these constituents, is clearly the operator of 
order 8 in question. 

When t is of order 4, G is the direct product of the tetra- 
hedral group and the cyclic group of order 4, since Si*, 52* are 
two operators of order 3 whose product is of order 2. When 
t is of order 2, G is the direct product of the tetrahedral group 
and the group of order 2, since {si^S2^)^ = Si^^ = l. Combining 
these results we arrive at the theorem : 

// two non-commutative operators s\, $2 satisfy the equations 
si^ = S2^, {siS2)- = \, they generate a group of order 96, or the direct 
product of the tetrahedral group and one of tite following three 
groups: tlie cyclic group of order 4, the group of order 2, tJie 
identity. 

Hence there are four and only 4 non-abelian groups which 
may be generated by two operators which satisfy these two 
conditions. The cyclic group of order 12 and its subgroups 
are evidently the only abelian groups which can be generated 
by two operators satisfying these conditions, and all of these 
groups can be generated by two such operators. 

Another generalization of the tetrahedral group is furnished 
by the equations 

Si^ = S2^, (5152)^ = 1. 

The cyclic group generated by s\^ = t is again invariant under 
G, and, as {s\-^S2S\-S2~^y = {siS2Si'^s\-^'^Y=Si-^^ is both in- 
variant under G and also trahsformed into its inverse by So, 
it results that the order of / is a divisor of 10. If we assume 
that / is of order 10, G is of order 120. We proceed to prove 
that this G is the direct product of the non-twelve group of 
order 24 and the group of order 5. If / is of order 10 the order 
of S\ may be assumed to be 30, since G would be abelian if the 
order of si were not divisible by 3. Hence we may assume 



154 GROUPS DEFINED ABSTRACTLY (Ch. VI 

that Si, 52 are of orders 30 and 20 respectively while Si'', $2^ 
are two operators of orders 6 and 4 respectively whose product 
is of order 6, since 



As (si^)^ = {s2^)^ = isi^S2^)^ is an operator of order 2, it results 
that (51*, 52*) is the non-twelve group of order 24, and that G 
is the direct product of this group and the group of order 5. 

When / is of order 5, G is evidently the direct product of the 
tetrahedral group and the group of order 5, and, when / is of 
order 2, G is the non-twelve group of order 24. This completes 
a proof of the theorem : 

// two non-commutative operators si, S2 satisfy the two condi- 
tions 5i^ = 52^, (5152)^ = 1, they must generate one of the following 
four groups: the tetrahedral group, the non-twelve group of order 
24, the direct products of these respective groups and the group of 
order 5. 

If 5i, 52 are commutative they must generate the group of 
order 15 or a subgroup of this group, and they may be so chosen 
as to generate any one of these subgroups. 

60. Generalization of the Octahedron Group. The equations 

5l2 = 52^ (5152)^ = 1 

evidently furnish a direct generalization of the octahedron 
group. To obtain an upper limit for the order of such a group 
we may consider the two operators 5i, 52~^5i52. They have a 
common square and this square is invariant under G. Hence 
it results that the commutator of S\, S2 is transformed into 
its inverse by each of the operators S\, 52~^5i52. Squaring this 
commutator there results 

5i - ^52 ~ ^5i52 5i - ^52 ~ ^5i52 = 5i - 85i52^5i525i52^5i52 

= 5i - 5i52^ 52 " ^5i - ^52 ~ *5i - * 525i52 = 5i - ^^5i525i52%l525i52 
= 5i-^'*5i525i525i-^52"^5i-^=5i -20 -52-^51 -^5251 
= 51-20. (5i-l52-^5i52)-^ 



601 GENERALIZATION OF THE OCTAHEDRON GROUP 155 

Hence it follows that {si-^S2-^siS2)^ = si-^^. As Si~^^ is trans- 
formed into its inverse by ^i, the order of si is a divisor of 40 
and the order of G must therefore be a divisor of 480. 

We begin with the case when Si is of order 8 and hence G 
is of order 96. To prove the existence of this G we may extend 
the non-twelve G24 by means of an operator of order 8 which 
transforms it according to an operator of order 2 correspond- 
ing to an outer isomorphism. Under the resulting group of 
order 96 the given G24 must therefore be transformed accord- 
ing to the octahedral group, which is the group of isomorphisms 
of this G24 as well as of its quaternion subgroup. If si' is the 
operator of order 2 in this octahedral group, which corresponds 
to the given operator Si of order 8 in the group of order 96, 
and if ^2' is an operator of order 3 in this octahedral group 
such that Si'so' is of order 4, we can select an operator 52 of order 
12 corresponding to $2' so that 51^ = 52^, (5iS2)* = l; for, all the 
operators of this Gge which correspond to operators of order 4 
in the octahedral group are of order 4, since the commutator 
subgroup of each of the three Sylow subgroups of order 32 is 
a subgroup of order 4 contained in the invariant quaternion 
subgroup. It is easy to verify that this group of order 96 is 
also generated by the following substitutions: 

Si = aa'gg'ee'cc' bo'hm'fk'di' id'oh'tnh'kf -jn'pl'nj'lp' , 
S2 = amlgkjeipcon bdfh a'm'l'g'k'j'e'i'p'c'o'n' h'd'J'h'. 

Having established the existence of a group of order 96 
generated by two operators of orders 8 and 12 which satisfy 
the given conditions, it is easy to find the group of order 480 
generated by two operators of orders 40 and 60 respectively 
which satisfy these conditions. In fact, it is obvious that 
such operators exist in the direct product of the given G96 and 
the cyclic group of order 5. For if 5i, 52 generate the former 
group and satisfy the relations si^ = S2^, {s\S2Y = \, and if / 
is an operator of order 5 which is commutative with each of 
these operators, then s\t, S2t~^ will also satisfy these equations. 
Hence we have proved that the largest group which can be 
generated by two operators which satisfy the two conditions 



156 GROUPS DEFINED ABiiTRACTLY (Ch. VI 

Si^=S2^, (si52y = l, is tjie direct product of the group of order 
5 and the group or order 9G obtained by extending the non- 
twelve group of order 21 by means of an operator of order 8 
which transforms it according to an outer isomorphism of 
order 2. 

It is evident that the group of order 48 obtained by estab- 
lishing a (2, 12) isomorphism between the cyclic group of order 
4 and the octahedron group is generated by two operators of 
orders 4 and 6 respectively which satisfy the given equations. 
Moreover, the direct product of this group of order 48 and the 
group of order 5, and the direct product of the octahedral 
group and this group of order 5 contain two generating opera- 
tors which satisfy the conditions under consideration. Hence 
we have arrived at the theorem: 

There are exactly six non-ahelian groups which can he generated 
by two operators which fulfil the equations 51^ = 52^, (5152)'* = 1. 
Three of these are of orders 24, 48, and 96, respectively, while 
the other three are the direct products of these respective groups and 
the group of order 5. 

A second generalization of the octahedron group is given 
by the equations 

SX^=S2\ (5152)2 = 1. 

Since the two operators 5i52, 525 1 are of order 2, they generate 
a dihedral group. To determine an upper limit of the order of 
this group we observe that 

(5251^52)^ = (52^5152^)^ -51^=5121. 

As sr^ is invariant under G, its order is a divisor of 42, and an 
upper limit of the order of this dihedral group is evidently 12, 
while the order of G is a divisor of 336. When s\ is of order 6 
the order of G is 48. Moreover, it is easy to see that the group 
of order 48, which may be obtained by extending the non- 
twelve group of order 24 by an operator of order 2 which trans- 
forms it according to an outer isomorphism, is generated by two 
operators of orders 6 and 8 respectively, which satisfy the given 
conditions. This group may be represented transitively on 
eight letters, and it involves 12 operators of each of the orders 



GO] GENERALIZATION OF THE OCTAHEDRON GROUP 157 

2 and 8 in addition to the given subgroup of order 24. It is 
the group of isomorphisms of the non-cyclic group of order 9. 

If we multiply the two given generators by an operator of 
order 7 and its inverse, the operator of order 7 being commuta- 
tive with each of these generators, we obtain two operators of 
orders 42 and 56 respectively which satisfy the conditions in 
question, and hence we have the theorem: 

// two operators satisfy the conditions Si^ = S2*, (5152)^ = 1, the 
largest group which they can generate is the direct product of the 
group of order 7 and the group of isomorphisms of the non-cyclic 
group of order 9. The total number of the non-ahelian groups 
which can he generated by two operators which satisfy these equa- 
tions is six: viz., the dihedral group of order 6, the octahedral 
group, the group of isomorphisms of the non-cyclic group of order 
9, and the direct products of these respective groups and the group 
of order 7. 

The third generalization of the octahedron group to be 
considered is given by the equation 

We may again consider the commutator of 5i, ^2 and observe 
that 

{Sl - ^S2 ~ ^SiS2)^ = (525152^)^51^ = 5i%2 -'^{s2^SiYs2^ 

= 5i652-2(5i452-^5i-0W = 5i*. 

As 5i^ is transformed into its inverse by 5i, it results that the 
order of 5i is a divisor of 36, and hence the order of G is a divisor 
of 432. It is easy to see that the group of order 48, which may 
be constructed by extending the non-twelve group of order 24 
by means of an operator of order 4 which has its square in this 
non-twelve group and transforms it according to an outer 
isomorphism of order 2, can be generated by two operators of 
orders 4 and 8 respectively which satisfy the given conditions. 
If we multiply this 5i and this 52 by an operator of order 9 and 
by its fifth power respectively, this operator of order 9 being 
commutative with each of the operators 5i, 52, and having only 
the identity in common with (51, 52), we obtain two operators 



158 GROUPS DEFINED ABSTRACTLY (Ch. VI 

of orders 36 and 72 respectively which satisfy the given con- 
dition and generate the group of order 432 in question. Hence 
it is easy to deduce the following theorem : 

I J two non-commutative operators satisfy the conditions Si^=S2*t 
(^1^2)^ = 1, they may generate the dihedral group of order 6, the 
octahedral group, the group of order 48 obtained by extending the 
non-twelve group of order 24 by means of an operator of order 4 
which has its square in this group and transforms it according to 
an outer isomorphism of order 2, the direct products of these respec- 
tive groups and a cyclic group of order 3 or 9. Hence there are 
exactly nine non-abelian groups which may he generated by two 
such operators. 

EXERCISES 

1. There are exactly six non-abelian groups whose two generators 
Si, St satisfy the equations Si^=Si^, (5152)^=1. They are the icosahedron 
group, a group of order 120, and the direct products of these respective 
groups and the cyclic groups of orders 5 and 25. 

2. If two commutative operators satisfy the equations 5i*=52*, 
(51^2)'= 1, they generate a cyclic group whose order is 3, 7, or 21; if they 
satisfy the equations Si^=S2^, (5152)^= 1, they generate a cyclic group whose 
order is 2, 4, 8, or 16; if they satisfy the equations Si^=Si^, (Ji5j)*=l, 
they generate a cyclic group whose order is either 5 or 25. 

3. There are exactly six non-abelian groups whose generators satisfy 
the equations Si^=Si^, {sis^^= 1. They are a group of order 1920 and the 
direct products of the icosahedral group and the cyclic group of order 2", 
a=0, 1, 2, 3, 4. 

4. If two commuiative operators satisfy the equation 5i*=^2S(5352)' = 1, 
they generate a cyclic group whose order is :; 3, 6, 9, or 18; if they 
satisfy the equations Si^=S2^, {siS^*=\, they geneiate a cyclic group whose 
order is 2, 4, 5, 10, or 20; if they satisfy the equations Si*=Si*, {siSiy=\, 
they generate a cyclic group whose order is 2, 7, or 14. 



CHAPTER VII 

ISOMORPHISMS 

61. Relative and Intrinsic Properties of the Operators of a 
Group. The operators or elements of a group have both 
relative and intrinsic properties. The latter relate to period- 
icity and are common to group operators and the roots of unity. 
Hence some of the earliest workers in abstract group theory 
associated this theory with the roots of unity. For instance, 
Cayley remarks that " a group may be considered as repre- 
senting a system of roots of the symbohc binomial equation 
5" = 1,"* and Sir W. R. Hamilton regards the icosahedral 
group as " a system of non-commutative roots of unity which 
are entirely distinct from the i, j, k of the quaternion though 
having some analogy thereto." f He calls this group the 
Icosian Calculus. 

In a non-abelian group the relative properties of the opera- 
tors are of greatest interest, while they are of comparatively 
little interest as regards abelian groups. In fact, they reduce 
to the question of common subgroups generated by these opera- 
tors in the latter case. In an automorphism of a group, the 
corresponding operators must evidently have the same intrinsic 
as well as the same relative properties. The great importance 
of the study of automorphisms rests on the fact that the proper- 
ties of an operator are the same as those of the operators which 
correspond to it in some autoniorphism of the group, and hence 
these properties need to be studied for only one of these corre- 
sponding operators. Thus the concept of isomorphisms econo- 
mizes thought, which is a fundamental object in mathematics. 

* Cayley, Philosophical Magazine, vol. 7 (1854), p. 40. 
t Hamilton, Philosophical Magazine, vol. 12 (1856) p. 446. 
' 159 



160 ISOMORPHISMS ICh. VII 

In its most elementary form the concept of isomorphisms 
is one of the oldest in mathematics, as it lies at the base of the 
development of abstract numbers. The concept of abstract 
number evidently rests on a kind of isomorphism between con- 
crete units of various kinds, so that for many purposes we 
may fix our attention entirely on what is common, viz., the 
abstract concept of units. In the theory of groups the con- 
cept of isomorphisms assumes a new importance in view of the 
fact that the different automorphisms of a group may be repre- 
sented by the corresponding substitutions on its operators, and, 
as was noted in 19, the totality of these substitutions con- 
stitute a group known as the group of isomorphisms,* or the 
group of automorphisms, of the original group. We have 
thus associated with each group its group of isomorphisms, 
which is of fundamental importance in many applications of 
the group. 

62. Group of Isomorphisms as a Substitution Group. If 
g distinct letters are placed in a (1, 1) correspondence with the 
operators of the group G.of order g, the symmetric substitution 
group will correspond to the totality of the possible arrange- 
ments of the operators of G. Such an arrangement cannot 
correspond to an automorphism of G unless the identity corre- 
sponds to itself. Hence the group of isomorphisms I oi G can 
always be represented as a substitution group on at most g 1 
letters, and its order must therefore be a divisor of (g 1)!. 
This order cannot be equal to {gl)\ except in caselpf three 
groups besides the identity, viz., the groups of orders ''2 and 3, 
and the four-group. In fact, it is evident that / cannot be more 
than doubly transitive on g 1 letters, since the correspondence 
of two operators fixes the correspondence of their product. 
In particular, the order of the group of isomorphisms of any 
finite group is a finite number. 

A necessary condition that / be transitive on g 1 letters 
is that all the operators of G besides the identity have the same 

Isomorphisms were first studied in an explicit manner by C. Jordan and 
A. Capelli. Their group properties were first studied by O. Holder and by E. H. 
Moore. 



62] ISOMORPHISMS REPRESENTED BY SUBSTITUTIONS 161 

order, and hence g must be of the form />", p being a prime 
number. Since the correspondence of two operators determines 
the correspondence of their powers, it is clear that / cannot be 
a primitive substitution group unless p = 2. If all the opera- 
tors of G besides the identity are of order 2, / is evidently 
doubly transitive. Hence it results that a necessary and suf- 
ficient condition that / be primitive on g l letters is that all 
the operators of G be of order 2.* 

The group / is generally intransitive on gl letters, and the 
number of its systems of intransitivity is equal to the number 
of complete sets of conjugate operators of G under /. In par- 
ticular, the number of characteristic operators of G is equal to 
g diminished by the degree of /. A sufficient condition that / 
is simply isomorphic with one of its transitive constituents, 
when it is represented as such a substitution group, is that 
G is generated by one of its complete sets of conjugates under /. 
When G is abelian it is generated by its operators of highest 
order, and these constitute a complete set of conjugates under /. 
As they constitute the only complete set of such conjugates 
that generate G, it results that the group of isomorphisms of an 
abelian group can always be represented in one and in only one 
way as a transitive substitution group on letters corresponding 
to operators of this abelian group. In other words, if the group 
of isomorphisms of an abelian group is represented on letters 
corresponding to the various operators of this abelian group, 
this group of isomorphisms has only one transitive constituent 
which is simply isomorphic with it, since every abelian group 
of order />*", except the group of order 2, admits a non-identical 
isomorphism in which every operator which is not of highest 
order corresponds to itself. A like theorem does not apply 
in general to the non-abelian. groups. In fact, if the / of a non- 
abelian group is represented on the ^1 letters corresponding 
to the operators of the group, excepting the identity, the 
number of its transitive constituents which are simply isomorphic 
with / may vary from zero to an indefinitely large number, as 
results from the alternating groups. 
* E, H. Moore, Bulletin of the American Mathematical Society, vol. 2 (189f), p. 33. 



162 ISOMORPHISMS (Ch. VII 

Suppose that G is abelian and that / is represented as a 
transitive group on letters corresponding to operators of G. 
These operators must be composed of the operators of highest 
order (m) in G. As any operator of highest order may be re- 
garded as an independent generator of G, it is evident that / 
cannot be regular unless G is cyclic and that / is always regu- 
lar when G is a cyclic group. As an abelian group cannot be 
represented as a non-regular transitive substitution group, it 
results directly from this fact that 

A necessary and sufficient condition that the group of iso- 
morphisms of an abelian group be abelian is that this abelian 
group be cyclic* 

The subgroup composed of all the substitutions of I which 
omit one letter must omit exactly <t){m) letters, <}>(m) being the 
totient of m, since an operator of order m generates <f>{m) oper- 
ators of this order. As the number of substitutions which are 
conlmutative with every substitution of a transitive group of 
degree n is a, where a is the exact number of the letters omitted 
by the subgroup of this transitive group which is composed of" 
all its substitutions which omit a particular letter, it results 
directly that / contains exactly <^(w) invariant substitutions, 
and that these substitutions transform every substitution of 
/ into the same power. Hence the theorem: 

// an abelian group involves operators of order m but none 
of higher order, its group of isomorphisms contains exactly ^{m) 
invariant operators. 

63. Groups of Isomorphisms of Non-abelian Groups. Sup- 
pose that G is non-abelian and that / is represented on letters 
corresponding to a set of operators of G. If all of these opera- 
tors were commutative they would generate a characteristic 
abelian subgroup of G. If this subgroup were in the central 
of G, I would involve operators corresponding to inner auto- 
morphisms, but which would not afifect any of the operators 
of the given set. This would also be the case if this subgroup 

* This theorem and the theorem of the following paragraph were proved 
abstractly in 41. The fundamental importance of these theorems seems to 
justify the present alternative proofs. 



5 63l ISOMORPHISMS OF NON-ABELIAN GROUPS 163 

were not in the central of G, since one of these operators would 
then transform into difTerent operators some of the operators 
of G which are not contained in this set. Hence it has been 
proved that 

The group of isomorphisms of a non-ahelian group cannot 
he represented on letters corresponding to a set of relatively com- 
mutative operators of this non-ahelian group. 

It is now easy to see that if the group of isomorphisms 7 
of any group G can be represented transitively on letters corre- 
sponding to a set of operators of G, then / cannot be more than 
doubly transitive. In fact, when G is non-abeUan and we fix 
the correspondence of two non-commutative operators of 
the set in an automorphism, the correspondence of a third 
operator has also been fixed, since such an operator may be 
obtained by transforming one of the two given operators by 
the other. That is, 

When the group of isomorphisms of any group G is represented 
on letters corresponding to operators of G, then this group of iso- 
'morphisms is at most douhly transitive. 

It is very easy to find non-abelian groups in which the / 
is doubly transitive if it is constructed in a given manner. As 
such a group we may consider the metacyclic group of degree 
p and of order p{p \). The / of this group can evidently be 
represented on the p letters corresponding to its operator of 
order 2, and if it is represented in this way it coincides with 
the metacyclic group itself. Instead of representing I on the 
operators of a group we may frequently represent it more con- 
veniently on a set of generating subgroups. If this is done 
I may be more than doubly transitive, as results from the fact 
that the group of isomorphisms of the tetrahedral group is the 
symmetric group of degree 4 if each letter corresponds to a sub- 
group of order 3. 

If / is a primitive group on letters corresponding to a set of 
operators of a non-abelian group G, its invariant subgroup 
corresponding to the inner isomorphisms of G must be transi- 
tive, since a primitive group cannot involve an intransitive in- 
variant subgroup. That is, if 7 is a primitive group on letters 



164 ISOMORPHISMS [Ch. VII 

corresponding to the operators of G, then G is either the abelian 
group of order 2" and of type (1,1, 1, . . . ) or G transforms 
transitively the set of operators to which the letters of / corre- 
spond. As every multiply transitive group is primitive, this 
theorem applies to all multiply transitive groups as well as to 
the simply transitive primitive groups. 

EXERCISES 

1. If G is the symmetric group of degree 4 its / may be represented as 
an imprimitive group of degree 6 on letters corresponding either to its 
operators of order 4, or to its six conjugate operators of order 2. These 
two imprimitive groups are not conjugate as substitution groups, since 
the one is composed of positive substitutions, while the other contains 
negative substitutions. 

2. The group of isomorphisms of the symmetric group of degree n, 
n^6, can be represented as a transitive substitution group on the n(n 1)/2 
letters corresponding to the transpositions of the symmetric group. When 
/ is thus represented, it is a simply transitive primitive group whenever 
n>4. 

Suggestions: Use the theorem that the symmetric group of degree 
n, ?^6 and n>2, is its own group of isomorphisms, and that it has no 
outer isomorphisms. See 65. 

3. Prove that if the / of the quaternion group is represented as a sub- 
stitution group whose letters correspond to its operators of order 4, it will 
be conjugate with the group in the first of these exercises which involves 
negative substitutions. 

64. Doubly Transitive Substitution Groups of Isomorphisms. 
If / is doubly transitive on letters corresponding to operators 
of G, each of these operators generates a cyclic subgroup (s) 
which is transformed into itself under the holomorph of G 
by a subgroup composed entirely of operators which are com- 
mutative with s; for, if a complete set of conjugate operators 
of G under / includes at least two powers of the same operator, 
the operators of this system must be transformed according 
to an imprimitive group. Suppose that si and S2 are two oper- 
ators of G which correspond to letters of I. We may assume 
that 5i, 52 are non-commutative; for, if all such operators were 
commutative, G would be abelian and hence the order of every 
operator of G would divide 2. Since this case is so elementary, 



64] DOUBLY TRANSITIVE GROUPS OF ISOMORPHISMS 165 

we shall exclude it in what follows and hence we shall assume 
that si, 52 are non-commutative. 

If si, 52 correspond to themselves in a given automorphism 
of G, all the operators of the subgroup generated by 5 1, 52 must 
also correspond to themselves and this subgroup must include 
more than two operators which are conjugate to 5i, 52 under /. 
Hence we have as a first result: 

// the group of isomorphisms of a group G can he represented 
as a doubly transitive group on letters corresponding to operators 
of G, then the subgroup composed of all substitutions which omit 
one letter of this doubly transitive group is eitlier imprimitive or 
it is a regular group of prime degree. 

This theorem follows directly from the well-known theorem 
that the subgroup which is composed of all the substitutions 
which omit one letter of a non-regular primitive group of degree 
n is always of degree w 1. When G is abeUan the given theorem 
evidently remains true and the imprimitive group in question 
involves systems of two letters each except when G is the four- 
group. 

When the subgroup which is composed of all the substitu- 
tions of / which omit one letter is a regular primitive group, 
the order of / is pip-^-l), p being a prime, and / involves p-{-\ 
subgroups of order p. It must therefore involve an invariant 
subgroup of order />+l which involves p conjugate operators 
under /. That is, the subgroup of order />+l must be the 
abelian group of order 2^ and of type (1,1,1,...). Hence 
the following theorem: 

// / is doubly transitive on letters corresponding to operators 
of G and if the subgroup composed of all the substitutions which 
omit one letter of I is primitive, then I is of order />(/>+!), P 
being a prime, and it involves an invariant subgroup of order 

p-\-l- 

When / is a doubly transitive group on letters corresponding 
to a set of conjugate operators of G, either all the operators of 
this set are commutative or no two of them are commutative. 
This results immediately from the fact that when / is doubly 
transitive any two of its letters can be transformed into an 



166 ISOMORPHISMS [Ch. VII 

arbitrary pair, but a pair corresponding to two commutative 
operators could not be transformed into a pair corresponding 
to two non-commutative ones. Hence it results that when 
/ is a doubly transitive group on letters corresponding to a set 
of operators of G, then G is either an abelian group of order 
2* and of type (1, 1, 1, . . . ) or no two of the operators of the 
set under consideration are commutative. 

65. Groups of Isomorphisms of the Alternating and the 
Symmetric Groups. In this section we propose to prove the 
theorems that the alternating and the symmetric group of 
degree n, W7^3, have the same group of isomorphisms, and 
that this group coincides with the symmetric group whenever 
w>3, with the exception of the single case when n = Q. In 
this special case the group of isomorphisms of the symmetric 
and alternating group is a well-known group whose order is 
1440; that is, this order is twice that of the corresponding 
symmetric group. The proof of these theorems entails the proof 
of several auxiliary theorems, which are also of considerable 
interest in themselves and of still greater historic interest in 
view of the fact that they relate to one of the oldest problems 
of group theory, viz., the determination of subgroups of small 
index under the symmetric and alternating groups. This is 
known as Bertrand's problem. 

As it will be desirable to use the theorems that the alternat- 
ing group of degree n, n9^Q, involves only one subgroup of 
mdex n, viz., the alternating group of degree n1, and that 
the symmetric group of degree n involves no subgroup of index 
, ?^6, besides the symmetric group of degree n1, we shall 
establish the somewhat more general theorem, sometimes called 
Bertrand's theorem,* that the symmetric group of degree n, 
n>4, has no subgroup whose index lies between n and 2, and 
that its only subgroups of index n, np^Q, are of degree w 1; 
moreover, the alternating group contains no subgroup of index 

Serret, Algibre sup^rieure, 1849, p. 267. Bertrand proved this theorem in 
1845, Journal de VEcole Poly technique, p. 129, by assuming the theorem, after- 
wards proved by Cebysev, that there is always at least one prime number be- 
tween /2 (exclusive) and n 2 (inclusive) whenever the natural number n 
exceeds 6. 



65] ISOMORPHISMS OF ALTERNATING GROUPS 167 

less than , >4, and its only subgroup of this index is the 
alternating group of degree n 1, when nj^Q. 

We begin with the proof of the latter part of this theorem, 
since the former part can be readily deduced from the latter. 
As the theorem of Cebysev, to which we have just referred, 
applies only to all numbers greater than 6, and the groups of 
degree seven are well known, we shall assume that n>7, and 
prove that the alternating group of degree n does not contain 
any subgroup whose index is less than +l, with the exception 
of its alternating subgroups of degree 1 and of index n. 
If such a subgroup existed it would be transitive on its letters, 
since the order of an intransitive subgroup could clearly not 
exceed 2'(n 2)!. As the order of an imprimitive subgroup is 
evidently less than this number, the subgroup in question would 
be primitive, and hence its order could not be divisible by the 
highest power of 3 which divides n!, since a primitive group 
of degree n does not involve a substitution of the form abc 
unless it is the alternating group of degree n. 

Since the order of the subgroup in question would not be 
divisible by the highest power of 3 that divides n\, this order 
would have to be divisible by the prime p, where n/2<p^n 2. 
Hence this subgroup would involve l-{-kp conjugate cyclic 
subgroups of order p. If two such subgroups had less than 
pl common letters, we could transform one by an operator 
of the other so as to obtain two such subgroups having a larger 
number of common letters without having all letters in common. 
This process could be repeated until two subgroups of order p 
would be found having p l common letters, and hence the 
primitive subgroup in question would itself involve primitive 
subgroups of each of the degrees p, p-\-l, ...,. It would 
therefore be at least four-fold transitive. 

As the transitive subgroup composed of all the substitutions 
involving a certain set of p letters would be invariant under 
a group of degree />+3, which would involve two transitive con- 
stituents of degrees p and 3 respectively, and as this transitive 
constituent of degree 3 would be the symmetric group of this 
degree, it results that each of the cyclic subgroups of order p 



168 ISOMORPHISMS (Ch.VII 

would be invariant under a group having the symmetric group 
of degree 3 as a trafisitive constituent. As the group of iso- 
morphisms of the group of order p is cyclic, this would imply 
that the subgroup, composed of all the substitutions in the 
primitive group of degree n and index less than w + l, which 
transform a subgroup of order p into itself, would involve 
substitutions of the form ah or dbc. As this is impossible, it has 
been proved that the alternating group of degree n, n>7, 
cannot involve a transitive subgroup of degree n and of index 
less than w+l. Hence the symmetric group of degree n,n>7, 
cannot contain a subgroup of index less than n-\-l and greater 
than 2 except the symmetric group of degree n1. 

From what precedes, it results that the only subgroups 
of index n in the symmetric and the alternating groups are those 
of degree n 1, whenever n>7. This theorem is known to 
be true also as regards the groups of degree 7. From this 
fact and from the theorem in 67, it follows directly that the 
group of isomorphisms of the alternating and of the symmetric 
group of degree n, w> 6, is the symmetric group of this degree. 

EXERCISES 

1. Prove that in an automorphism of the symmetric group of degree 
6 substitutions of the form abc may correspond to those of the form abc-def, 
and that all the operators of order 3 in this symmetric group are conju- 
gate under its holomorph. 

2. Prove that the symmetric groups of degrees 4 and 5 are complete 
groups, and that the alternating groups of these degrees have the corre- 
sjxjnding symmetric groups for their groups of isomorphisms. 

3. Give an instance of a group which involves an invariant subgroup 
whose group of isomorphisms is larger than that of the entire group. 

66. Several Useful Theorems Relating to the Groups of Iso- 
morphisms.* Every abelian group can be extended so that we 
obtain a group of twice the order of the original group, by 
means of operators of order 2 which transform each operator 
of this abeUan group into its inverse. These groups may be 
regarded as a direct generalization of the dihedral groups, and 
may therefore be called generalized dihedral groups as regards 
Cf. PhUosophical Magazine, vol. 231 (1908), p. 223. 



5 661 SEVERAL USEFUL THEOREMS 169 

the given abelian groups. If the given abelian subgroup involves 
operators whose order exceeds 2, the corresponding general 
dihedral group is non-abelian and vice versa. Let G be any 
non-abelian generalized dihedral group of order g and let H 
be the abelian subgroup of order g/2 which was extended to 
obtain G. In any automorphism of G the g/2 non-invariant 
operators of order 2 must correspond to themselves, and hence 
the I oi G can be represented as a substitution group of degree 
h, h being the order of H. 

It is evident that the non-invariant operators of order 2 
in G can be arranged in h different ways after the automorph- 
ism of H has been fixed. Hence the order of the / of G is 
the same as the order of the holomorph of H. We proceed 
to prove that the / of G is simply isomorphic with the holo- 
morph oi H. In fact, this / can be represented as a transitive 
substitution group of degree // which involves an invariant 
regular subgroup which is simply isomorphic with H, since G 
can be made simply isomorphic with itself in such a way that 
the operators of H correspond to themselves while the remain- 
ing operators of G correspond to themselves multiplied by an 
arbitrary operator of H. These isomorphisms therefore corre- 
spond to a regular subgroup of order h in /, / being represented 
on letters corresponding to the non-invariant operators of order 
2 in G, and this regular subgroup is simply isomorphic with H 
by construction. 

Moreover this regular subgroup is invariant under /, since 
H must correspond to itself in every automorphism of G, and 
this regular subgroup includes- all the substitutions of / corre- 
sponding to the automorphisms of G in which all the opera- 
tors of H correspond to themselves. From this fact it results 
that / must be a subgroup of the holomorph of H, and as the 
order of / is equal to the order of the holomorph of H it results 
that / is this holomorph. These results may be stated as 
follows: 

The group of isomorphisms of the generalized dihedral group 
of an abelian group H, involving operators whose orders exceed 2, 
is the holomorph of H. 



170 ISOMORPHISMS [Ch. VII 

As a special case of this theorem we may observe that the 
group of isomorphisms of the dihedral group of order 2h, h> 2, 
is the holomorph of the cyclic group of order h. 

If E is any abelian group of even order, it may be extended 
by means of h operators of order 4 such that they have a common 
square, and each operator of H is transformed into its inverse 
by each of these operators of order 4. The group G of order 
2// which can be constructed in this way will be called the 
generalized dicyclic group as regards H, since it reduces to the 
dicyclic group whenever H is cyclic. With the single exception 
when H is of order 2^ and of type (2, 1, 1, 1, ... ), the I of 
G can always be represented as a transitive substitution group 
on the given h operators of order 4. By exactly the same 
reasoning as was employed in the preceding case we see that, 
when E does not satisfy the given special condition, the I of 
this G is also the holomorph of G whenever the common square 
of the given h operators of order 4 is a characteristic opera- 
tor of E. This proves the following theorem: 

The group of isomorphisms of the generalized dicyclic group 
as regards an abelian group E, which is not both of order 2" and 
type {2, 1, 1, 1, . . . ), is the holomorph of this abelian group 
whenever the common square of the h operators of order 4 which 
were added to E is a characteristic operator of E. 

It should be observed that i7 is a characteristic subgroup of 
G also when it is both of order 2" and of type (2, 1, 1, 1, . . . ) 
provided the squares of the remaining operators of order 4 
in G are not the same as those of the operators of order 4 in E. 
In this case, as well as in the more general case considered 
above, the group of isomorphisms of G is the subgroup of the 
holomorph of E composed of all the operators of this holomorph 
which are commutative with the square of the given h operators 
of order 4. The method of proof employed above may serve 
to establish a very elementary but useful theorem, which may 
be stated as follows: 

// a group G containing a characteristic subgroup E is such 
that automorphisms of G may be obtained by multiplying suc- 
cessively an operator s of G which is not in E by all the operators 



661 SEVERAL USEFUL THEOREMS 171 

of H while the operators of H are left unchanged, then the group 
of isomorphisms of G involves an invariant subgroup which is 
simply isomorphic with 11, whenever this group of isomorphisms 
can be represented as a transitive group of degree h, corresponding 
to the conjugates of s. 

Suppose that G is a complete group which involves only 
one subgroup of index 2, and consider the direct product of G 
and the group of order 2. If a group contains only one sub- 
group of index 2, this subgroup is generated by the square of 
the operators of the group, and, conversely, if a subgroup of 
index 2 is generated by the squares of the operators of a 
group, it is the only subgroup of index 2 in the group. That 
is, a necessary and sufficient condition that a group contain 
one and only one subgroup of index 2 is that the squares of its 
operators generate such a subgroup. Hence the squares of the 
operators in the direct product of G and the group of order 
2 generate a characteristic subgroup of index 4 under this 
direct product. The / of this product involves an invariant 
operator of order 2 corresponding to the automorphisms in 
which two of the three co-sets as to the given characteristic 
subgroup are multiplied by the invariant operator of order 2. 
As the order of this / is the double of the order of G and as / 
involves an invariant operator of order 2 which is not in the / 
of G there results the theorem: 

If G is a complete group and contains only one subgroup of 
index 2, then the group of isomorphisms of the direct product of G 
and the group of order 2 is simply isomorphic with this direct 
product. 

As special cases of this theorem we may observe that the / 
of the direct product of the symmetric group of degree n, n> 2 
and 5^6, and the group of order 2, is simply isomorphic with 
this direct product; and that the / of the direct product of the 
metacyclic group of order p(p l), p being an odd prime, 
and the group of order 2, is simply isomorphic with this direct 
product. If a group G is the direct product of characteristic 
subgroups, the / of G is evidently the direct product of the 
/'s of these subgroups. As an abehan group is the direct prod- 



172 ISOMORPHISMS [Ch. VII 

uct of its Sylow subgroups, the I of an abelian group is always 
the direct product of the Vs of its Sylow subgroups. 

67. Group of Isomorphisms of a Transitive Substitution 
Group. Suppose that G is a transitive substitution group of 
degree n which involves no subgroups of index n and degree , 
but involves subgroups of degree w 1. Its subgroups of 
degree 1 must therefore correspond among themselves 
in every automorphism of G, and these subgroups may be so 
lettered that they are transformed by every substitution in G 
in exactly the same manner as the letters of this substitution 
are transformed. From this it results that if each of the n 
subgroups corresponds to itself in any automorphism of G, each 
of the substitutions of G must also correspond to itself in this 
automorphism. That is, the I oi G may be represented on 
letters corresponding to these subgroups. 

As G involves subgroups of degree w 1 , it is simply isomorphic 
with its group of inner isomorphisms. Hence the I oi G may be 
represented as a transitive substitution group of degree n which 
contains G invariantly. This proves the following theorem: 

// G is a transitive substitution group of degree n which involves 
subgroups of degree n 1 but no subgroups of both degree n and 
index n, then the group of isomorphisms of G can be represented 
as a transitive substitution group of degree n which contains G 
as an invariant subgroup. 

As the symmetric group of degree n involves no subgroup 
of degree and index n when w?^6, and as it contains a subgroup 
of degree n l whenever w?^2, it results from the given theorem 
that the I of every symmetric group of degree w, except when 
n is either 2 or 6, can be represented as a substitution group 
on n letters, which contains this symmetric group. This sub- 
stitution group must therefore be the corresponding sym- 
metric group, as was proved above. In a similar way we may 
observe by means of this theorem that the metacycUc group 
of degree p a.nd of order p{p \) is its own group of isomorph- 
isms. These illustrations may suffice to point out the use- 
fulness of this theorem in the study of the groups of isomorphisms 
of substitution groups. 



S 67] ISOMORPHISMS OF TRANSITIVE GROUPS 173 

When G is an intransitive group of degree n such that every 
subgroup which omits one letter is of degree exactly n 1, 
it is still true that these 1 subgroups of G are transformed 
by every substitution of G in exactly the same manner as the 
letters of this substitution are permuted. If G is such that 
these n subgroups of degree w 1 must correspond to them- 
selves in every possible automorphism of G, the group of iso- 
morphisms of G can again be represented as a substitution group 
on n letters. It is, however, not necessary that this substitu- 
tion group should be transitive, as may be seen by letting G 
represent the intransitive group of degree 7 and of order 24 
obtained by establishing a (1, 4) isomorphism between the sym- 
metric groups of degrees 3 and 4. 

EXERCISES 

1. There is no group whose group of isomorphisms is a cyclic group of 
odd order greater than 1.* 

2. The quaternion group and the cyclic group of order 8 are the only 
two groups of this order that cannot be the groups of isomorphisms of any 
group, t 

3. There are two and only two groups which have the symmetric groups 
of order 6 for their group of isomorphisms. J 

4. Find the groups of isomorphisms of all the substitution groups whose 
degrees do not exceed 5. 

5. The order of the group of isomorphisms of any abehan group is 
divisible by the number of its operators of highest order. A necessary 
and sufl5cient condition that the order be equal to this number is that the 
group be cyclic. 

6. The number of distinct groups which have a given group of inner 
isomorphisms is either zero or infinity. 

7. If a non-abelian group can be represented transitively only as a 
regular group, it cannot be the group of isomorphisms of an abelian group. 

* Annals of Mathematics, second series, vol. 2 (1900), p. 79. 

t Bulletin of the American Mathematical Society, vol. 6 (1900), p. 339. 

X Transactions of the American Mathematical Society , vd. 1 (1900), p. 399. 



CHAPTER Vm 
SOLVABLE GROUPS* 

68. Introduction. A group is said to be solvable if, and 
only if, it contains a series of invariant subgroups such that the 
last of the series is the identity and the index of each of these 
subgroups under the next larger subgroup is a prime number. 
For instance, the symmetric group of order 24 contains an 
invariant subgroup of index 2, this subgroup contains an invari- 
ant subgroup of index 3, this subgroup is the four-group and 
contains an invariant subgroup of index 2, and the identity is 
of index 2 under this last subgroup. Hence the symmetric 
group of order 24 is solvable. The numbers 2, 3, 2, 2 are said 
to be its factors of composition. In general, the factors of 
composition of a group are the indices of the successive largest 
invariant subgroups. For example, the symmetric group of 
order 120 contains an invariant subgroup of index 2, but this 
subgroup involves no invariant subgroup besides the identity. 
Hence the symmetric group of order 120 is insolvahle and has 
2 and 60 for its factors of composition. Every abelian group is 
evidently solvable. 

The terms solvable and insolvable as applied to groups of 
finite order are transferred from the theory of equations. An 
algebraic equation is solvable by rational processes in addition 
to root extractions whenever the group of the equation is solv- 
able and only then (cf. Part III). It should be observed that 
an invariant subgroup of an invariant subgroup is not neces- 
sarily an invariant subgroup of the entire group. For instance, 
the invariant subgroup of order 2 used in connection with the 

* In H. Weber's Lehrbuch dcr Algebra, solvable groups are called melacydic. 
In the present work we use the term metacyclic with its older meaning to repre- 
sent the holomorph of the group of order p. See p. 12. 

174 



68l INTRODUCTION TO SOLVABLE GROUPS 175 

symmetric group of order 24 in the preceding paragraph is not 
invariant under this symmetric group, but it is invariant under 
the four-group. 

It may appear possible that one might obtain only prime 
factors of composition by one method of selecting the succes- 
sive invariant subgroups while another method would lead to 
composite factors. If this were possible the determination of 
solvability or insolvability of a group would sometimes require 
an examination of different sets of subgroups such that each 
is a largest invariant subgroup of the one which precedes it, 
and the last is the identity. That the factors of composi- 
tion of a group are entirely independent of the order in which 
the invariant subgroups in question are selected can easily 
be established by means of the theorem that two invariant 
subgroups which have only the identity in common must 
have the property that each operator of the one is commuta- 
tive with every operator of the other (cf. 29). We proceed 
to prove the invariance of the factors of composition of any 
group by means of this theorem. 

Let Go be any solvable group, and let the following series 
of subgroups, with the exception of Go, have the property 
that each is invariant and of prime index under the one which 
immediately precedes it: 

Go, Gi, G2, , Gp-i, Gp=l. 

In selecting another such series, suppose that the first a groups 
coincide with the first a groups in given series, but that the 
(a-f l)th is different. We thus have the series: 

Go, Gi, . . . , Ga-l, G a, . . . 

As both Ga and G'a are invariant under Ga-i, their cross-cut 
is also invariant under Ga-i- The quotient group of Ga-i 
with respect to this cross-cut must therefore involve two 
maximal invariant subgroups, corresponding to Ga and G'^, 
which have only the identity in common. Hence this quotient 
group is the direct product of these invariant subgroups of 
prime orders. This proves that the given cross-cut could be 



176 SOLVABLE GROUPS (Ch. VIII 

selected for Ga+i regardless of whether Ga or G'a was selected, 
and hence it establishes the invariance of the factors oj com- 
position of any solvable group, since this invariance may be 
assumed for the groups G^ and G', as they are of lower order 
than Go is. 

It may be observed that the above proof is not dependent 
on the fact that Go was assumed to be solvable. The quotient 
group with respect to a given cross-cut is always the direct 
product of two simple groups, since the invariant subgroups 
are maximal, and hence the factors of composition are invari- 
ants as regards any group, even if this group be insolvable. 
Moreover, as the two factor groups of a direct product of two 
simple groups are evidently invariants of this direct product,* 
it results immediately that the following series of quotient 
groups 

Go/Gi, G1/G2, . , Gp-i/Gp = Gp-\ 

is an invariant of Go irrespective of whether Go is solvable or 
not. That is, the totality of these simple quotient groups is 
independent of the choice of the maximal invariant subgroups. 
The important theorem as regards the invariance of the factors 
of composition of any group was first proved by C. Jordan 
in Journal de M athematiques , volume 14 (1869), page 139. The 
fact that the given series of quotient groups is also an invariant 
of Go was observed by O. Holder in the M athematische An- 
nalen, volume 34 (1889), page 37. 

Instead of dej&ning a solvable group as one having only 
prime factors of composition, we may also dejQne it as a group 
which has the property that we arrive at the identity by form- 
ing the successive commutator subgroups. That is, if Ga 
is the commutator subgroup of Ga-i, a = l, 2, . . . , and if we 
form the series of groups 

Go, Gi, G2, . . . , Gx, 

* This is a special case of the theorem that factor groups of any direct product 
are always completely determined by this direct product. Cf. Remak, Crelle's 
Journal, vol. 139 (1911), p. 293. 



69] SERIES OF COMPOSITION 177 

a necessary and sufficient condition that Go be solvable is that 
for a finite value of X, Gx = l. It is evident that this implies 
that Gx-i is abelian and that the order of Ga is less than that 
of Ga-i whenever a^X. The given condition for the solva- 
bility of Go is therefore equivalent to saying that a necessary 
and suflUcient condition that a group be solvable is that none 
of its successive commutator subgroups besides the identity 
is a perfect group (cf. 29). 

That this second definition of a solvable group is equivalent 
to the first, follows immediately from the fact that if a group 
has an invariant subgroup of prime index, this subgroup must 
include the commutator subgroup of the group, and if the 
order of the commutator subgroup of a group is less than the 
order of the group, there must be an invariant subgroup of prime 
index in the group, since the commutator quotient group is 
always abelian. 

While every simple group of composite order is evidently 
a perfect group, there are perfect groups which are composite. 

EXERCISES 

1. The smallest perfect group which is not also simple is of order 120. 

2. The factors of composition of the symmetric group of degree , 
n5^4, are 2 and !/2. 

3. Everj' perfect group besides the identity is insolvable, but an insolv- 
able group is not necessarily perfect. 

4. Every subgroup of a solvable group is solvable. 

5. Each one of the series of successive commutator subgroups is invari- 
ant under the original group. 

6. Every solvable group of composite order contains an invariant 
subgroup which is abelian and whose order exceeds unity. 

69. Series of Composition. If each one of the series of groups 

(A) Go, Gi, G2i . . . , Gp = l, 

excluding the first, is a maximal invariant subgroup of the one 
which immediately precedes it, the series is said to be an 
ordinary series of composition of Go. For brevity an ordinary 
series of composition is often called a series of composition. 
A necessary and sufficient condition that Go be a simple group 



178 SOLVABLE GROUPS (Ch.VIII 

whose order exceeds unity is that this series consists of only 
two terms. We may form another series 

(B) Go, G I, G 2, . , G\ = l 

in which each subgroup is an invariant subgroup of Go and has 
the property that no larger invariant subgroup of Go, con- 
taining this one, exists in the group which immediately pre- 
cedes it. The series {B) is said to be a chief series of composi- 
tion of Go. It is sometimes possible to select a series of com- 
position in such a manner that it is also a chief series. This 
can evidently always be done when the order of Go is a power 
of a prime number. 

It is always possible to construct an ordinary series of com- 
position by inserting some terms in a chief series of composi- 
tion if the chief series is not already an ordinary series of com- 
position. Suppose that it is necessary to insert some terms in 
the series (B) between G'^ and G'a+i to obtain an ordinary series 
of composition and that Hi is such a term, which corresponds 
to a maximal invariant subgroup in the quotient group 
G'a/G'a+i, while H2 is a conjugate of Hi under Go. Since 
Hi and H2 are maximal invariant subgroups of G' their cross- 
cut is also invariant under G'a, and the corresponding quotient 
group is the direct product of two conjugate simple groups. 
When Go is solvable these simple groups have the same prime 
order p, and hence this quotient group is of order p^. 

As every group of order p^ is abelian, it results that the 
cross-cut of Hi and any of its conjugates under Go involves 
the pth power of every operator in these conjugate subgroups 
as well as the commutators of all their operators. Hence it 
follows that if we find the complete set of conjugates of Hi 
under the solvable group Go, their common cross-cut, which is 
invariant under Go and hence coincides with G'a+i, involves 
all their commutators as well as the pth powers of all their opera- 
tors. This proves the following theorem: 

// Go is any solvable group and Go, G/, G'2, , G'\ = l 
is a chief series of composition, then the quotient group of any 
of the groups in this series with respect to the one immediately 



69) SERIES OF COMPOSITION 179 

following it is an ahelian group which involves only operators 
of prime order besides the identity. 

This abelian group is therefore of type (1, 1, 1, ... ). 
If the order of this quotient group is p", we must evidently 
insert a conjugates of Hi in order to obtain an ordinary series 
of composition from the given chief series. The first of these 
can be chosen in (/>" !)/(/> 1) different ways. 

When Go is insolvable, the given method of proof leads di- 
rectly to the results that the quotient group of any one of the 
groups, in the given chief series, with respect to the one immedi- 
ately following it, is a direct product of simple groups of com- 
posite order which are simply isomorphic. If these simply 
isomorphic simple groups are of prime order, we have the result 
expressed in the preceding theorem. It is easy to prove that 
the totality of the quotient groups of each group of a chief 
series of composition with respect to the one following it is an 
invariant of the group. In fact this proof is practically the 
same as the proof of the fact that the factors of composition 
of any group is an invariant of the group. 

The theorem that the quotient group of any group in a chief 
series of composition, with respect to the one which follows it, 
is a power of a simple group, results also directly from the fact 
that this quotient group cannot involve a characteristic sub- 
group. The given method of proof leads directly to the theorem 
that a necessary and sufficient condition that a group does not 
contain a characteristic subgroup is that this group is a power 
of a simple group. If this theorem had been assumed as known, 
the fact that each of the given quotient groups is a power of a 
simple group would not have required any proof. 

If we form the successive commutator subgroups of a solv- 
able group Go we obtain a third series 

(C) Go, G I, G 2, , G y = \. 

As the quotient group of each of these groups with respect to 
the one which immediately follows it is abelian and as each one 
of the successive commutator subgroups is invariant under 
Gt), it results that as regards the three series A, B, C we have 



180 SOLVABLE GROUPS [Ch. VIII 

p>\>y. We may clearly pass from the successive commuta- 
tor subgroups series to a chief series of composition by insert- 
ing additional terms wherever necessary. It should be ob- 
served that in the successive commutator subgroups series all 
the terms of the series are invariants of Go, and hence the order 
of the quotient group of each term of the series with respect 
to the one just following it is also an invariant of Go, while in 
series A and B we proved only that the totality of these quo- 
tient groups is an invariant of Go, but the order in which these 
two quotient groups occur is not necessarily an invariant in 
these two series. 

To illustrate the difference between the series A, B, C, 
we first use for Go any group of order p'^, m>2. It is clear 
in this special case that p = \ = m, while 7<w/2 when m is 
even and 7<(w+l)/2 when m is odd. This results directly 
from the fact that every group of order p~ is abelian. If Go 
is the symmetric group of order 24 it results that p = 4 while 
X = -y = 3. A necessary and sufficient condition that Go be abelian 
is that 7 = 1. Although it is always possible to pass from a 
successive commutator subgroups series to a chief series of 
composition by inserting (if necessary) terms into the former 
series, it is not always possible to obtain the commutator sub- 
groups series by dropping terms out of a chief series of composi- 
tion. Similar remarks apply to series A and B\ that is, it is 
not always possible to obtain a chief series of composition by 
omitting terms of an ordinary series of composition, but it is 
always possible to obtain an ordinary series of composition 
by inserting terms, if necessary, between terms of a chief series 
of composition. 

The fact that it is not always possible to obtain a chief 
series of composition by omitting some groups of an ordinary 
series which is not also a chief series, should be emphasized, 
since erroneous statements in regard to this matter are rather 
common. For instance, such erroneous statements appear 
in the Encyclopedie des Sciences Malhematiques, tome 1, 
volume 1, page 568. As a very simple illustration it may be 
observed that if G is the octic group, and if Gi is a non-cyclic 



70] INVOLVING ONE NON-CYCLIC 8YL0W SUBGROUP 181 

group of order 4 contained in G, we. may construct an ordinary 
series of composition of G by using for G2 a subgroup of order 
2 contained in Gi but not invariant under G. By omitting Gi 
from this series there results a series which is not a chief series 
of composition of G, since Gi contains the commutator subgroup 
of G. The fact that a commutator subgroup series of compo- 
sition cannot always be obtained by omitting subgroups from 
a chief series which is not also a commutator subgroup series 
can be illustrated by means of the direct product of the octic 
group and a group of order 2. 

A solvable group of composite order must be composite, but 
not every composite group is solvable. Sometimes the proof 
that all the groups which belong to a certain system are com- 
posite is equivalent to the proof that they are solvable. This 
is clearly the case when the invariant subgroups and the cor- 
responding quotient groups belong to the same system. For 
instance, the proof that every group whose order is the product 
of distinct prime numbers is composite is equivalent to the 
proof that all such groups are solvable. Similarly, the proof 
that every group whose order is a power of a prime is com- 
posite is equivalent to the proof that all such groups are solv- 
able. On the contrar}', the proof that every group whose order 
is divisible by 2 but not by 4 is composite does not estabUsh 
the fact that such a group is solvable. If it could be proved 
that every group of odd order is composite, it would result 
from this that every group whose order is not divisible by 4 
would be solvable. 

70. Groups Involving no More than one Non-cyclic Sylow 
Subgroup. One of the most useful theorems as regards solv- 
able groups is the one which afl5rms that a group is solvable 
if it involves either no non-cyclic Sylow subgroup or contains 
only cychc Sylow subgroups 'besides those whose orders are 
divisible by the highest prime which divides the order of the 
group. To prove this theorem we assume that the order of such 
a group G is written in the form g = pi'^p2'^ . />x"^, where 
p\, p2, - - , p\ ^rt distinct prime numbers, arranged in 
ascending order of magnitude. 



182 SOLVABLE GROUPS [Ch. VIII 

Since G involves operators of orders />i"", the number of its 
operators whose orders divide g/pi is less than g. This number 
is known to be a multiple of g/pi ( 25) and hence it can be 
written in the form kg/ pi, where k is an integer. The number 
of operators of G whose orders are divisible by pi"^ is there- 
fore equal to 

g-kg/pi=l{pi-l), 

since this number is also a multiple of the number of the differ- 
ent possible generators of a cyclic subgroup of order pi"^. The 
first member of the given equation is divisible by g/pi, and, 
as each of the prime factors of this divisor exceeds pi I, it 
results that / is also divisible by g/pi. Hence k = l, and 

I=g/Pi- 

If ai> 1, it can be proved, in exactly the same way, that the 
number of the operators of G whose orders are divisible by 
/>!" ^ but not by pi"' is 

g/pi-kig/pi^=liipi-l). 

Hence ^i = l, and li=g/pi^- By continuing this process it 
results that the number of operators of G whose orders divide 
Pfi'fi . P\">', /3<X, is exactly equal to this number. In par- 
ticular, G contains only one Sylow subgroup of order px'x, 
and the corresponding quotient group contains only one sub- 
group of order ^x_i"x-i, etc. Hence G contains a cyclic quo- 
tient group of order />i"', and the invariant subgroup of G 
which corresponds to the identity in this quotient group is 
such that each of its Sylow subgroups, with the possible 
exception of those of order />xx, is cyclic. This completes a 
proof of the following theorem: 

// all the Sylow subgroups whose orders are not divisible by the 
highest prime which divides the order of the group are cyclic, the 
group is solvable. 

In particular, every group whose order is not divisible by 
the square of a prime number is solvable. Hence there is only 
one such group when none of these primes diminished by unity 
is divisible by another. 



711 SUCCESSIVE GROUPS OF INNER ISOMORPHISMS 183 

71. Groups whose nth Group of Inner Isomorphisms is the 
Identity. Let 

h, /2, . . . , / 

represent a series of successive groups of inner isomorphisms 
of a group G. A necessary and sufficient condition that /i 
be simply isomorphic with G is that the central of G be the 
identity. In this case, all of the successive groups of inner 
isomorphisms are simply isomorphic with G. If / has the 
same order as la+i, a<n, then / is simply isomorphic with all 
the groups of the given series which succeed /. It may happen 
that for a sufficiently large value of n, / = !. In this case G 
is clearly solvable. 

Suppose that one of the prime numbers which divide the 
order of G does not divide the order of /. From the theorem 
that a group has exactly the same number of Sylow subgroups 
of any order as its group of inner isomorphisms has, where the 
identity is counted as a Sylow subgroup of order p" if the order 
of the group of inner isomorphisms is not divisible by p, it 
results that G involves only one Sylow subgroup of order />"* 
whenever the order of /, for a sufficiently large value of n, is 
not divisible by p. 

If G is a solvable group whose order is divisible by p, while 
the order of / is not divisible by p, then G is a direct product 
of its Sylow subgroups of order p"* and some other subgroup. 
On the other hand, it is evident that when G is such a direct 
product, then it is possible to find a number n such that the 
order of / is not divisible by p. Hence the theorem: 

A necessary and sufficient condition that a solvable group G 
be the direct product of a Sylow subgroup of order p^ and some 
other subgroup is that the order of the nth group of inner iso- 
morphisms of G should not ' be divisible by p when n is sufficiently 
large. 

In particular, a necessary and sufficient condition that G 
be the direct product of its Sylow subgroups is that we can 
arrive at the identity by forming successive groups of inner 
isomorphisms of G* 

* Cf. Loewy, Malhematische AnnaUn, vol. 55 (1901), p. 69. 



184 SOLVABLE GROUPS [Ch. VIII 

EXERCISES 

1. If the th succcssfve group of inner isomorphisms of a group is the 
identity, the wth successive group of inner isomorphisms of each of its 
subgroups must also be the identity. 

2. A group of order 455 is necessarily cyclic. 

3. The direct product of any finite number of solvable groups is again 
a solvable group. 

4. If n is the number of operators or subgroups in a complete set of 
conjugates of a simple group, the order of this group divides n\ and is a 
multiple of n. 

72. Arbitrary Choice of Factors of Composition. If a group 
G is the direct product of its Sylow subgroups, then each sub- 
group of G is also such a direct product, since we must arrive at 
the identity by forming successive groups of inner isomorphisms. 
Hence it results directly that it is possible to find a series of 
composition of G such that the factors of composition occur in an 
arbitrary order, whenever G is the direct product of its Sylow 
subgroups. 

On the other hand, suppose that G is a solvable group such 
that it is possible to select a series of composition in such a 
manner that all the factors of composition which are equal to 
a given prime number p appear at the end of the series, while 
all the other factors of composition are prime to this number. 
Hence G contains one and only one Sylow subgroup of order />". 
In particular, there results the theorem: 

A necessary and sufficient condition that a series of composi- 
tion of a solvable group G can be found which corresponds to an 
arbitrary arrangement of the factors of composition is that G 
is the direct product of its Sylow subgroups. 

By combining the theorem of the present section with those 
of the preceding one, it results that a necessary and sufficient 
condition that a series of composition of a solvable group can be 
found such that the corresponding factors of composition occur 
in an arbitrary order, is that we can arrive at the identity by 
forming successive groups of inner isomorphisms. These results 
can readily be extended so as to apply to groups which have 
composite factors of composition. Such a group is also the 
direct product of groups whose orders are powers of given 



73] GROUPS OF ORDER ?>V 185 

factors of composition whenever there is a series of comp)osi- 
tion corresponding to every possible order of the factors of com- 
position. 

73. Groups of Order p" g'', p and q being Prime Numbers. 
In 138 it will be proved that every group of order p''(f is 
solvable. Two very simple special cases will be considered 
here. The case when /3 = 1 is especially simple and will be 
considered first. If G is of order p'^q and q<Pj then G contains 
only one subgroup of order />" and it must therefore be solvable. 
When q> p and G contains more than one subgroup of order 
/>", it must contain just q such subgroups. We proceed to prove 
that no two of these subgroups can have a cross-cut whose 
order exceeds unity. 

Let K represent the largest possible cross-cut of a pair of 
these subgroups. Since K is invariant under operators of each 
of these subgroups which are not contained in K, it must be 
invariant under a subgroup of G whose order is divisible by 
some prime number besides p. Hence K is invariant under an 
operator of order q, and it is therefore contained in each one 
of the q subgroups of order p". Since K is composed of all the 
operators which are common to a complete set of conjugates, 
it is invariant under G, and the corresponding quotient group 
has an order which is of the same form as the order of G.. 
It remains therefore only to consider the case when the q sub- 
groups of order P" are such that no two of them have two 
operators in common. 

In this case these q subgroups are transformed according 
to a transitive substitution group of degree q which involves no 
substitution whose degree is less than ^1. It can therefore 
not contain more than ^1 substitutions of degree q, since 
the average number of letters in all its substitutions is qh 
Hence it contains only one subgroup of order g,,and it must 
therefore be composite. As no group of order p'q can be 
simple, every group whose order is of this form must be solvable. 

The case when all the Sylow subgroups of a group of order 
pq^ are abelian is almost equally elementary. Let G be such 
a group and suppose that G is simple. If s represents any 



186 SOLVABLE GROUPS [Ch. VIU 

operator of order p contained in G and if J is any Sylow subgroup 
of order ^, it is evident that 5 can be transformed into all its 
conjugates under G by means of the operators of J. If each 
operator of G is represented by a substitution according to which 
this operator transforms the conjugates of s, there results a 
substitution group S which is simply isomorphic with G, since 
G is simple. 

To the subgroup 7 in G there corresponds a transitive sub- 
stitution group in 5, since J transforms s into all its conju- 
gates. As a transitive abelian group is regular, s has exactly 
c^ conjugates under G. Hence the identity is the cross-cut 
of any two of the (f Sylow subgroups of order p" contained in 
G. That is, if G were simple it would contain (/?" 1)^ opera- 
tors whose orders are powers of p, and hence it could contain 
only (f operators whose orders are powers of q. As such a 
group could contain only one subgroup of order ^, it could not 
be simple. This proves that every group of order p^cf is solv- 
able whenever the Sylow subgroups of orders p" and q^ are 
abelian. 

74. Insolvable Groups of Low Composite Orders. From the 
theorems which have been established it follows directly that 
every group whose order is less than 60 is solvable. That 
there is an insolvable group of order 60 and that this is the lowest 
order of a simple group of composite order was observed by 
E. Galois. We proceed to prove that there is only one insolv- 
able group of this order. If a group of order 60 contains only 
one subgroup of order 5, the corresponding quotient group is 
of order 12 and hence the group is solvable. Hence an insolv- 
able group of order 60 must involve 6 conjugate subgroups of 
order 5 and must transform them according to a transitive sub- 
stitution group of degree 6 and order 60. As there is only one 
such substitution group,* there is only one insolvable group of 
order 60. 

* The fact that there could not be more than one such substitution group may 
be seen as follows: Such a group contains the group of degree 5 and of order 10, 
and hence it involves exactly 15 substitutions of order 2. As none of these can 
occur in two subgroups of order 4 the group must contain five subgroups of this 
order. 



74] INSOLVABLE GROUPS OF LOW COMPOSITE ORDERS 187 

The first order beyond 60 which is not included in the given 
theorems is 72. As this is of the form S/?"*, p being an odd prime 
number, we shall prove that no group whose order is of this 
form is insolvable. If such a group were insolvable it would 
contain 4 or 8 subgroups of order />"*, and hence p would be 
3 or 7. In the former case the 4 subgroups of order f* would 
be permuted either according to the symmetric or according 
to the alternating group of degree 4. In either case there would 
be a solvable invariant subgroup and the corresponding quotient 
group would also be solvable. Hence the entire group would be 
solvable. 

If there were 8 subgroups of order p^, then p would be 7, 
and the 8 subgroups of order />"* would be permuted according 
to the group of degree 8 and of order 56. As this group is 
solvable, it results that no group whose order is of the form 
8/>" can be insolvable. 

Similarly it may be observed that no group whose order is of 
form 4/>'" can be insolvable. The only remaining number less 
than 100 which requires consideration is 84. Every group of 
order 84 involves an invariant subgroup of order 7. Hence 
such a group is solvable, and there is one and only one insolvable 
group whose order is less than 100. As there is no number be- 
tween 99 and 120 which could be the order of an insolvable 
group we proceed to consider the insolvable groups of 
order 120. 

Since a group of order 120 involves either one or six sub- 
groups of order 5, every insolvable group of this order contains 
6 subgroups of order 5, which it transforms according 
to a transitive group of degree 6. Every group of order 
120 which contains six subgroups of order 5 is insolvable, 
since the order of the group according to which these six 
subgroups are transformed is either 60 or 120. If this 
order is 120 the group in question must be simply iso- 
morphic with it, and hence it must be simply isomorphic 
with the symmetric group of degree 5. If the order of this 
group of degree 6 is 60, the groups in question must have a 
(2, 1) isomorphism with the icosahedral group and must 



188 SOLVABLE GROUPS [Cm Mil 

involve two substitutions (51, $2) which satisfy one of the 
following two sets of conditions: 

In the former case si , 52 generate the icosahedral group, and 
the group in question is the direct product of this group and the 
group of order 2. In the latter case si, S2 generate the group 
of order 120, which has no subgroup of order 60, but has the 
icosahedral group for a quotient group. This proves that 
there cannot be more than three insolvable groups of order 120. 
As it is well known that three such groups exist, it has been 
proved that there are only three insolvable groups of order 120; 
viz., the symmetric group of this order, the direct product of the 
icosahedral group and the group of order 2, and the group of order 
120 which involves operators of order 4 and has the icosahedral 
group for a quotient group. 

Between 120 and 168 are four numbers which are not 
included in the general theorems which were established above. 
These numbers are 132, 140, 144, and 156. If a group of any 
of these orders were insolvable it would have to be simple. 
Hence we may confine ourselves to a proof of the theorem 
that every group of these orders must be composite. If 
a group of order 132 were simple, it would permute its 
twelve subgroups of order 11 according to a simply iso- 
morphic transitive substitution group of degree 12. As this 
would be of class 11 it would contain an invariant subgroup 
of order 12. 

A group of order 140 = 22- 5- 7 contains a characteristic 
subgroup of order 5, a group of order 144 contains 1, 3, or 9 
subgroups of order 16, and hence cannot be simple, and a group 
of order 156 = 22- 3 -13 contains a characteristic subgroup 
of order 13. Hence 168 is the lowest order, beyond 120, of an 
insolvable group. We proceed to prove that there is only 
one simple group, and hence only one insolvable group, of this 
order. That there is at least one such group results from the 
theory of the transitive substitution groups of degree 7, and 



74) INSOLVABLE GROUPS OF LOW COMPOSITE ORDERS 189 

from the group of isomorphisms of the abelian group of order 
8 and of type (1, 1, 1). 

A simple group of order 168 contains 8 subgroups of order 
7, and can therefore be represented as a transitive substitution 
group G of degree 8. A maximal subgroup of G is of degree 
7 and of order 21, and it therefore involves seven subgroups of 
the form (abc-def). Each of these subgroups is transformed 
into itself by six substitutions under G. Hence it may be 
assumed that all the possible simple groups of order 168 con- 
tain a particular subgroup of degree 7 and order 21, and are 
generated by this subgroup and a substitution of the form 
ab-cd-ef-gh, which transforms into itself a particular subgroup 
of the form (abc-def) contained in the given subgroup of 
order 21. 

If the given subgroup of order 6 is the symmetric group 
of this order, the three possible substitutions of the form 
ab-cd-ef-gh are completely determined by the subgroup of 
the form (abc-def). That is, there is not more than one 
transitive group of degree 8 which contains a particular sub- 
group of degree 7 and of order 21, and which is such that 
its six substitutions which transform into itself a particular 
subgroup of the form (abc-def) constitute the symmetric 
group of order 6. It is clear that there is one such group, 
since the simple group of degree 7 and of order 168 trans- 
forms its eight subgroups of order 7 according to a transitive 
group of degree 8. 

To prove that there is only one simple group of order 168 
it remains only to prove that a transitive group of degree 8 and 
of order 168 cannot be simple if the subgroup of order 6 which 
transforms a subgroup of the form (abc-def) into itself is 
cyclic. In fact, such a transitive group contains 28 cyclic 
subgroups of order 6, and hence it contains 48 -|- 56-1-56 sub- 
stitutions of orders 7, 3, and 6 respectively. It can therefore 
contain only one subgroup of order 8. Hence we have com- 
pleted a proof of the theorem that iJtere is only one simple group 
of order 168, and hence there is also only one insolvable group of 
this order. 



190 SOLVABLE GROUPS (Ch. VIII 

The only order between 168 and 200 which is not included 
in the general theorems which have been established is 180, 
and it is known that there is at least one insolvable group of 
this order, viz., the direct product of the icosahedral group and 
the group of order 3. We proceed to prove that this is the 
only insolvable group of order 180. A group of this order 
must have one, six, or thirty-six subgroups of order 5. If 
it has only one such subgroup it is evidently solvable. It 
could not contain thirty-six such subgroups, since these sub- 
groups would form a single set of conjugates. As an operator 
of order 2 could not have 45 conjugates, the group would 
contain operators of order 6. As the number of its 
operators of each of the orders 3 and 6 would be even and 
a multiple of 5, and as it would contain just 36 operators 
whose orders are prime to 5, it results that such a group 
would involve not more than 5 subgroups of order 3. Hence 
there is no group of order 180 which contains 36 subgroups 
of order 5. 

It remains to consider the possible groups of order 180 which 
contain exactly six subgroups of order 5. In this case the groups 
must be isomorphic with the icosahedral group on six letters 
and they must therefore involve an invariant subgroup of 
order 3. If we can prove that such a group G must also con- 
tain the icosahedral group invariantly, it must be the direct 
product of the icosahedral group and the group of order 3, 
and hence there is only one such group. If the subgroups of 
order 9 ' in G are non-cyclic, it is evident from the given 
isomorphism that G contains two operators, si, S2, which 
satisfy the conditions Si^ = S2^ = {siS2y = l, and hence G con- 
tains the icosahedral group invariantly. It remains there- 
fore only to prove that the subgroups of order 9 in G cannot 
be cyclic. 

If these subgroups were cyclic, G would clearly contain two 
operators /i, /2 which would satisfy the conditions /i^ = /2^ = 
{iihy, /i = l. Hence the equations 

ti =t2tlt2, t2 =tit2tl. 



74) INSOLVABLE GROUPS OF LOW COMPOSITE ORDERS 191 
If we consider the powers of the commutator 

we have 

(tit2trU2-^y=tr%t2^ti%%=ti-%t2Hit2% 

{ht2h -H2-^f=h- ^tlt2Hit2HiH2% =tr ^tit2%t2%t2Hl 

= h- Hit2HxH2hH2Hi = /i - Hit2Hxt2Hit2Hi 
=h-niH2hH2HiH2h^ = h-HiH2HiHi^h^ 
(hhh -U2-^y = tr ^HiH2HiH2ni^ ht2Hit2Hx = tx^ 

As txt2tx~^t2~^ is the product of t\t2 and /l-^/2~^ and as t\t2 
and /2/1 have a common square, it results that tx^ is transformed 
into its inverse by /1/2. Since it is also invariant under /1/2, 
the order of tx cannot exceed 6; but this order divides 9 and 
hence it must be 3. That is, G involves the icosahedral group, 
and hence there is only one insolvahle group of order 180, viz.^ 
the direct product of the group of order 3 and the icosahedral group. 
We have now considered all the possible insolvable groups 
whose orders are not greater than 200 and found that there 
are six such groups, vjz., one of each of the orders 60, 168, 
180, and three of order 120. From the simplicity of the above 
considerations it appears probable that these enumerations 
could readily be carried much further, but enough may have 
been done to exhibit the general nature of the problems involved. 
Several years ago Holder carried this investigation through all 
groups whose orders are less than 480 with the results exhibited 
in the following table: 



Insolvable Groups * 



Order 60 

Number of groups. .... 1 



120 


168 


180 


240 


300 


336 


360 420 


3 


1 


1 


8 


1 


3 


6 1 



Holder, Mathematische Annalen, vol. 46 (1895), p. 420. 



192 SOLVABLE GROUPS [Ch. VIII 

EXERCISES 

1. A group whose Sylow subgroups of order 2*" are cyclic contains 
an invariant subgroup of index 2. 

Suggestion: Represent the group as a regular group. 

2. If a primitive substitution group of degree n is solvable, n must 
be a power of a prime and the primitive group must be contained in the 
holomorph of the abelian group of order n and of type (1,1,1, . . . ). 

Suggestion: Consider a chief series of composition of the primitive 
substitution group. 

3. If a transitive group of the prime degree p is solvable, its order is a 
divisor of />(/> 1). 



I 



PART II* 

FINITE GROUPS OF LINEAR HOMOGENEOUS TRANS- 
FORMATIONS 



CHAPTER IX 

PRELIMINARY THEOREMS 

Linear Transformations, 75-82 

75. Introduction and Definition. It is often of importance 
in analysis to exchange one set of variables for another, the 
variables of either set being linear homogeneous functions of 
the variables of the other set (cf. Ch. XVIII), as in coordinate 
geometry : 

x = x' cos Qy' sin Q. 

(1) 

y=%' sin Q-\-y' cos Q. 

We assume that a function /(:*;, y) is given, in which the new 
variables {x\ y') are to be put in place of the old (x, y) by means 
of (1) ; this is called operating upon f by the linear transforma- 
tion (l). 

A capital letter is in general used to denote a linear trans- 
formation; thus, we shall here denote (1) by S. The result 
of operating upon f{x, y) by S may then be indicated sym- 
bolically as follows: 

(2) (f)S=f{x' cos e-y' sin d, x' sin B-\-y' cos d). 

* This part was written by H. F. Blichfeldt. 
193 



194 



LINEAR TRANSFORMATIONS 



[Ch. IX 



However, since/ is generally subjected to several transforma- 
tions successively, we find it convenient after each operation 
to drop the accents. In the future we shall therefore write, 
instead of (2), 

(f)S=f{x cos 6y sin 6, x sin d+y cos $). 

Formal Definition. A linear transformation in n vari- 
ables x\,X2, . . , Xnis a set of linear homogeneous equations 

Xi=aiix\-\-ai2x'2-\- . . . -\-ainx'ni 



A: 



Xn='an\X'\^-an2x'2+ +aT', 



expressing the original variables x\, . . . , x^ in terms of new 
variables x'l, . . . , x'n, under the condition that the equations 
can be solved for the latter. 

76. Matrix and Determinant. It is customary to represent 
the transformation A above by the matrix of the coefficients 
(called the matrix of A): 



or simply 



an ai2 . . . ai 






*nn. 



The determinant of this matrix is called the determinant of A . 

77. Inverse of a Linear Transformation. The solutions of 
the equations A ( 75) for a;'i, . . . ,:', say 

x'i=buXi-^h\2X2-\- . . . +&in^, x'2 = etc., etc., 

will constitute a new linear transformation which will take the 
conventional form adopted for A after the accented and unac- 
cented variables have been interchanged: 

Xi=bnx'\-\-bi2x'2-\- . . . -\-binx'n, 



B: 



Xn=bnlX\+bn2x'2-\- +&nn^'. 

This is called the inverse of A, and is denoted by ^4"^ (cf.;! 22). 



78) PRODUCT OF LINEAR TRANSFORMATIONS 196 

78. Product of Linear Transformations. If a function / be 
subjected to two linear transformations successively, A and B, 
the result is equivalent to operating upon / by a single linear 
transformation C, called the product of A and B. We shall prove 

Theorem 1. The proditct oj two linear transformations 
in the same variables, 

A=[a,t], B = [bJ, 
is a linear transformation C = [f,J, where 

n 

Csi = a,ibu-\-agJ)2t-{- . . . -\-asabt= x^flw^a; 

symbolically, AB = C. 

Proof. Consider a function f{xi, . . . , Xn) operated upon 
by^: 

(f)A=fiyi, . . . ,y), 
where ( 75) 



)>$= y ^asyXf,. 



The result of operating upon (f)A by B, namely {{f)A)B, which 
we shall write (f)AB, is then/(2i, . . . , z), where 

n n n ^ n . n 

Accordingly, if C be the linear transformation defined above, 
we have 

(f)C=fizi, . . . , Zn) = (f)AB. 

Note that the element c,i in the product -45 = C is obtained 
by multiplying the elements of the 5th row of A into the corre- 
sponding elements of the /th column of B and adding the results. 

79. The Commutative and Associative Laws. The com- 
mutative law does not hold in general; that is, C=AB and 
C'=BA do not represent the same linear transformation. On 
the other hand, the associative law holds always: A{BC) 
{AB)C whatever be the transformations A, B and C; that is, 
if we write S for the product BC and T for AB, then AS = TC 



196 



LINEAR TRANSFORMATIONS 



[Ch.- IX 



This we prove by comparing the matrices of these products, 
as obtained by the application of Theorem 1. 

80. Canonical Form of a Linear Transformation. Identical 
and Similarity-transformations. A transformation whose matrix 
has zero elements everywhere except in the principal diagonal 
is said to have the canonical form (or to be written in the canonical 
form) : 



Xl=a\X I, X2'=C12X 2, 



Xn ClnX n 



In such a case we employ the notation S = {<xi, a2, . . . , ai^. 

If the coefficients ai, a2, , an, which are called the 
multipliers of S, are all equal, we say that 5 is a similarity- 
transformation; if they are all equal to unity, 5 is the identical 
transformation or the identity. Denoting the latter by E and 
any transformation by ^, we have EA=AE=A. 

81. Power and Order of a Linear Transformation. Since 
the associative law holds for a product, it follows that we may 
write A^ for AA, A^ for {AA)A, etc., and call these products 
the second, third, etc., powers of A. Moreover, denoting the 
inverse of -4" by A~^, we have A'*A~'' = A~'*A" = E, and 
A~'* = {A-^y. The index laws hold for positive and negative 
integral powers if we interpret A^ as E. 

Usually no power of a linear transformation A taken at 
random will be the identity. If, however, such a power exists, 
we say that A is oi finite order, and the lowest power of A which 
equals the identity is called the order oi A. 

EXERCISES 

1. Prove that the determinant of the product of two transformations 
A and B is equal to the product of the determinants of A and B. Hence 
prove that the determinant of A is the reciprocal of that of ^4"^ 

2. Find the inverse of S, 75. 



3. Prove that if 
A = 



B= 



P q 

r s 
then B is the inverse of A . 

4. Construct AB and BA, where 



s 
r 



B= 



-9 

P} 



ps-qr=l, 



[S 



83l 



LINEAR GROUP 



197 



5. Find the general form of a linear transformation in three variables 
which is commutative with 



5= 



a 











a 





,0 





b, 



ay^b. 



6. Prove that a similarity-transformation is commutative with any 
linear transformation in the same variables. 

7. Prove that the multipliers of a transformation of finite order are 
roots of unity (cf. 116). 

82. Remarks. For the purpose of avoiding a possible confusion on 
the part of the student of the terms " literal substitution " and " linear 
substitution," the term " Unear transformation " has here been adopted 
throughout. 

A different value for Cst from that given in 78 is obtained by inter- 
preting the linear transformations A and 5 in a different manner or by 
writing last in the product that transformation which operates first, as 
is the custom with functional operators. Thus, Klein, Jordan and Burn- 
side regard the variables in the left-hand members of A, 75, as the 
new, and those in the right-hand members as the old variables (the 
accents being placed accordingly or entirely absent), and therefore get 
Csj=2"_i0iw^to instead of the value given in 78; while Weber, attach- 
ing the same meaning to the linear transformations, inverts the order in 
the product, writing BA where the authors mentioned write AB. On 
the other hand, Frobenius and Schur, though writing ys for x's in the 
equations A, 75, interpret linear transformations in the same way and 
arrive at the same results as the author of this Part II. The latter has 
deviated from his customary notation in papers published on the subject 
to the extent of dropping accents of variables that were formerly sup- 
plied with them and vice versa. 



Groups of Linear Transformations, 83-87 

83. Linear Group. If to is an imaginary cube root of unity, 
the six transformations 





1 


Ai = 




[0 Ij 




w2 


^4 = 






,w 0, 





1 


A2 = 


> 




1 0, 




V 


As = 






.0 w, 





a 





^3 = 








.0 


<^. 







0) 


Ae = 








2 


0. 



2 3 


, T = 


"4 6' 


1. 




,0 2. 



198 GROUPS OF LINEAR TRANSFORMATIONS (Ch. IX 

form a group of order 6 which is isomorphic with the sym- 
metric group in 3 letters (cf. 1). Here Ai is the identical 
transformation; the inverse of Az is A5, while A2, A^. and Aq 
are their own inverses. The transformations ^3 and As are both 
of order 3; and ^2, ^4, ^6 all of order 2. 

A set of g distinct linear transformations Si, . . . , Sg will 
form a group of order g if the conditions of 22 are all fulfilled. 
Such a group we shall simply call a linear group. 

84. Collineations and Collineation-groups. It is often con- 
venient not to regard as essentially different two transforma- 
tions whose matrices can be obtained one from the other by 
multiplying all the elements of one by a constant factor, as, for 
instance, in the case of 



S = 



The two transformations are then said to represent the same 
collineation. In other words, a collineation is specified by the 
mutual ratios of the elements of the corresponding matrix, 
not by the actual values of these elements. In practice it is 
customary to affix a factor of proportionality to either the old 
or the new variables to distinguish a collineation from a linear 
transformation; the collineation represented by 5 or T above 
would thus be written 

pxi = 2a;'i -\-Zx'2, pX2 = X'2' 

If A and B are two distinct linear transformations which 
represent the same collineation, then BA~^=A-^B is a simi- 
larity-transformation. For, let the common ratio of the ele- 
ments of the matrix of B to the corresponding elements of the 
matrix of A be 6, then we immediately verify that B =AS=SA, 
where 5 = (0, 6, . . . , 6). 

If now a linear group G of order g be giVen, two cases may 
arise. Either no two distinct transformations of G will repre- 
sent the same collineation, or there will be some one set of, 
say /, transformations which all represent the same collinea- 
tion. In the former case G contains g distinct collineations; 



841 



COLLINEATION-GROUP 



199 



in the latter, the g transformations of G can be arranged into 
g/f sets, of / transformations each, furnishing g/f distinct col- 
lineations. For, let i4i, . . . , ^4/ be any set of transforma- 
tions representing the same collineation, then 



A2Ar^=S2, 



, AfAr^=Sf 



are similarity-transformations and are all distinct. Hence, 
if the group F of similarity-transformations Si, S2, . . . , S/ 
contained in G (cf. Ex. 1, 87) is of order /, we have /=/. 
On the other hand, if A is an arbitrary transformation in G, 
then the / distinct transformations -451, , ^5/ all repre- 
sent the same colUneation, so that/'=/. Hence /'=/. 
The sets of G can therefore be exhibited as follows: 

Si, S2, . . , Sf', 

ASi, AS2, . . , AS/; 

BSi, BS2, . . , BSf] 



To each line will correspond a single collineation. Moreover, 
if the product of a transformation from a set (a) and a trans- 
formation from a set (/3) fall in the set (7), then the product 
of any transformation from (a) and any transformation from 
(/3) will fall in (7), since the two products merely differ by a 
similarity-transformation. Accordingly, the group G is (/, 1)- 
isomorphic with an abstract group E of order h=g/J, namely 
the quotient-group G/F (13). Since a collineation in n 
variables can be interpreted as a projective transformation in 
space of 1 dimensions by using homogeneous coordinates, 
the abstract group H becomes a group of operators of order 
h, called the collineation-group corresponding to G. 

85. An Example. Take the linear group G of order 8: 



.4i = 


1 
.0 1. 


> 


A2 = 




1 
-1 0. 


, ^3 = 


-1 
- 




1, 


,A4- 


= 




,1 


-1 




Bi = 


^1 

.0 - 




1, 


,B2- 


= 


1 

1 0, 


,Bs = 


-1 
. 1. 


> 


^4 = 






1 


-1 

0. 



200 



GROUPS OF LINEAR TRANSFORMATIONS (Ch. IX 



The group F of similarity-translormations is here of order 2: 
A I and ^43. The table of 84 will therefore consist of four 
lines as follows: 



Ai, 


As] 




^1, 


A3\ 


A2A1, 


A2A3; 




^2, 


^4; 


BiAi, 


B1A3; 




Bu 


53; 


B2A1, 


B2Ar, . 




B2, 


54. 



The corresponding collineation-group, of order 4, can be written 
by selecting a representative from each of the four lines, as 
A\, A2, Bi, B2. 

Remark. It will be noticed that in this example neither 
the set Ai, A2, Bi, B2, nor any other set that may be selected 
from the four lines will form a linear group of order 4. 

86. Groups of Linear Transformations of Determinant Unity. 
It follows from the manner in which the product AB oi two 
Hnear transformations A, B is formed (78) that the deter- 
minant Dab of ^5 is the product of the determinants of A and B : 

Dab = DaDb. 

Now let ,5 be a transformation belonging to a collineation- 
group G of order g in w variables: 

A, B, . . . , o, . . . , 

and Ds the determinant of S. A linear group isomorphic with 
G and whose transformations have unity for the value of their 
determinants may then be constructed in the following manner. 
Let be a solution of the equation 

(3) e'' = Ds-\ 

and let S\ be the similarity-transformation {d, 6, . . . , 6), 
the value of whose determinant is 6". Then S2 = SSi is of 
determinant unity. Since (3) has n solutions, there are n 
different transformations associated with 5 in this manner, 
say 52"', . . , S2^^^- Hence the following correspondence: 

G: A, ... 5, . . . 



G2: 



/1 2 , . . . , yi2 



,(1) 



. . , 5: 



.(a). 



5 871 



UNEAR FRACTIONAL GROUP 



201 



It is easy to verify that the set of transformations in the last 
line form a linear group (Go) which is (w, l)-isomorphic with 
the collineation-group G; in fact, if AB = C, and a, /3 arbitrary 
accents, then 

where 7 is fully determined. 

As an illustration, let G be the collineation-group 



^ = (1, 1), B = 
We find 

^i< = (l, l),^i^ = (-l, -1); 5i^" = (i ), 5i^^' = (-|, -h), 
and therefore 



G2: 



^2"^ = (1, 1), 

-1 



52"> = 







^2<^> = (-i, -1); 
1 



^2^2) = 



-1 Oj 



It may, however, be possible to find a subgroup of G2 of 
lower order whose corresponding collineation-group is Ukewise 
G, and whose transformations also have unity for the value 
of their determinants (cf. 110). This will be the case if G 
is a group of odd order in two variables ( 97). When in the 
future we mention a group of linear transformations of deter- 
minant unity corresponding to a given collineation-group or 
linear group, we shall mean such a group G3 of lowest possible 
order, having the same collineation-group as that given or as 
that corresponding to the given linear group. 

87. Linear Fractional Group. When only the mutual 
ratios of the elements of the matrices are of importance and not 
their actual values, we may adopt another mode of representing 
the operators, namely by writing them in linear fractional 
form. Let a given linear transformation be 



A: 



x, = a,ix'i + 



+ax', (j = l, 2,-. . . ,if), 



202 GROUPS OF LINEAR TRANSFORMATIONS (Ch. IX 

and let the ratios xx/xn, , ^n-iA be denoted by yi, 
. . . ,yn-\ respectively.' Then from A we get 

y,= S ; ; ^ 1 U=i, 2, . . . ,n-i). 

ai>'i+ -rann-iyn-i-ronn 

To a similarity-transformation here corresponds the identity 

ys = y', (5 = 1, 2, . . . , n-1), 

and we see that the linear fractional group is simply isomorphic 
with the corresponding collineation-group and may be regarded 
as its equivalent. 

EXERCISES 

1. Prove that the similarity-transformations contained in a linear 
group G form a subgroup which is invariant in G. 

2. Prove that the determinant of a linear transformation belonging 
to a finite group is a root of unity (cf. 116). 

3. Among the determinants of the transformations of a group G of 
order g let there be one which is a root of unity whose index is the power of 
a prime p (116). Prove that if all those transformations be eliminated 
whose determinants contain as a factor a root whose index is the highest 
power of p occurring among such indices, then will the remaining trans- 
formations form an invariant subgroup of G of order g/p. 

In particiJar, prove that the transformations of determinant unity 
form an invariant subgroup of G. 

4. Let T be one of the transformations of the group in the last exer- 
cise whose determinant contains a factor 6 of index />", and assume that 
p is relatively prime to the number of variables n. Then we can always 
find a root of unity, say ^, of index />", whose nth power is e~ ^ If now all 
the elements of the matrix of T be multiplied by ^, the determinant of the 
new transformation T' will no longer contain as a factor a root whose index 
is a power of p. 

, Dt= w. Here ^= w, and T'= 
wj 

Now prove that if all the transformations be modified in this manner 

we shall obtain a group G' which is isomorphic with G. 

Evidently, to a possible similarity-transformation (a, a, ... , a) 

of G whose multipliers a are roots of unity of index a power of p will 

correspond the identity (1, 1, ... , 1) of G'. In such a case therefore 

the order of G' will be that of G divided by a power of p. 

5. Construct the group Gj corresponding to the linear group of order 
8 given in 85. 



For instance, let T 



(o :)) 



88l CHANGE OF VARIABLES 203 

6, The group listed in 83 is of order 6. Show that the corresponding 
coUineat ion-group and group d are of orders 6 and 12 respectively. 

7. Construct the collineation-group and group Gj corresponding to 
the linear fractional group of order 6: 

y=y', W, 1-y', l/(l-y'). (y'-l)/y', y7iy'-l). 

88. Change of Variables. Before subjecting a given func- 
tion /(x, y) to a linear transformation or a group of linear trans- 
formations, we may introduce a different set of variables in 
the function and in the transformations. We shall examine 
in detail the important case where the new variables are linear 
homogeneous functions of the old; in other words, where the 
new value of / is obtained from the old by subjecting the latter 
to a linear transformation T which expresses the change of 
variables considered. 

To illustrate, let 5 be the transformation 

S: x = x' cos 6y' sin 6, y = x' sin d-\-y^ cos 6. 
We now suppose that new variables X, Y are introduced, where 

T: :r=i(X+F), y=l.(X-Y), 

and correspondingly 

r: x'=\{r-\-Y'), y=l.(r-FO, 

where P= 1. The function / becomes 

U)T-f{\{X-^Y), 1(X-F))=F(X, F), say, 

and the transformation S expressed in the new variables (Si) 
is found by solving for X, Y from the equations 

|(X+F) =^(r +F0 cos 0-l(X'-Y') sin d, 
l{X-Y)=l(r-hY') sin 0-\-l.(X'-Y') cos d. 



204 GROUPS OF LINEAR TRANSFORMATIONS [Ch. IX 

We obtain 

Si: X^X'^', Y = Y'e-*r 

The function F may now be subjected to the transformation 
S\, producing 

(4) {F)S, = {(J)T)S,=^if)TSi. 

Obviously, the final result could equally well have been 
obtained by operating first upon f hy S before introducing the 
new variables; that is, the final expression in (4) is also obtained 
by introducing X', Y' in 

{f)S=f{x' cos d-y sin e, x' sin 0+/ cos Q) 

by means of T', giving 

({f)s)r=(f)sr. 

Hence we have 

{f)TSi = (J)ST', 

so that, since / is an arbitrary function, 

TSi=sr, 

or 

Si = T-^ST'. 

As remarked in 75, we drop accents after operating by a 
linear transformation. Accordingly, our final formula b 

(5) Si = T-'ST. 

In the general case, the same symbolic result is obtained. 
Hence the 

Theorem 2. Let there be given a linear group 

G: A, B, . . . , 

and a change of variables in the form of a linear transformation 
T; then we obtain in the new variables a linear group 

Gi: T-^AT, T-^BT, .... 

These two groups are simply isomorphic (13, 24), and we shall 
write the latter symbolically T'^GT. 



CHARACTERISTIC EQUATION 



205 



89. Characteristic Equation. If we add ^ to each of the 
elements in the principal diagonal of the matrix of a linear 
transformation A = [a^] and equate the resulting determinant 
to zero, we have an equation in 6 which is called the character- 
istic equation of A: 



(6) 



ail 6 ai2 

fl21 ^22 & 



a2n 



Onl 



On2 



Onn-d 



= 0. 



Theorem 3. If T and A be linear transformations, the roots 
of the characteristic equation of A are the same as those of T~ M T. 

To prove this theorem, let us put T~^AT=B = [b,t], whose 
characteristic equation is 



(7) 



bii-d . 



bin 



bnl 



= 0. 



Regarding as a variable temporarily, we denote the transfor- 
mations whose matrices are the left-hand members of (6) and 
(7) by A-d and B-d. Then, since T" M T = B, and T-^ST = S, 
where 5 is the similarity-transformation (6, 6, . . . , 6), we 
may readily prove that T~^(Ad)T = B d. Hence, if the 
determinants of T,A d and B dhe denoted by p, q, r, we have 
(cf. Ex. 1, 81) p~^qp = r, so that q = r. Accordingly, the 
coefficients of the various powers of in ^ and r are equal, and 
the theorem follows. 

The sum of the characteristic roots of A is called the char- 
acteristic oi A. It is equal to the sum of the elements in the 
principal diagonal of A, namely aii+a22+ . -\-chut' 

EXERCISES 

1. Prove that if S, 88, is a similarity-transformation, then Si=S. 
Prove also that if S and T both have the canonical form, then Sx=S. 

2. Prove (5) in the case of two variables directly by multiplying out 
the right-hand member (cf. 77, 78). 

3. Find the characteristic roots and characteristic of a transforma- 
tion written in canonical form; also of the transformation (1), 75. 



206 



GROUPS OF LINEAR TRANSFORMATIONS [Ch. IX 



90. Transitive and Intransitive Groups. Consider a group 
G in four variables whose transformations all have the typical 
form 

b 



A = 



a I 
g c 
h d 



If in this group we introduce new in variables yi, y2, 2i, Z2 such 
that yi=xi-{-X2, y2 = X3-\-X4; zi=xiX2, Z2 = X3X4, the trans- 
formations take the simpler form 



Ai = 



It seems natural to adopt the notation 

{A' 0' 



p 


q 








r 


s 














/ 


u 








V 


w 



Ai = 



A" 



A' representing a matrix in yi, y2 only, and A" one in zi, Z2 
only. Correspondingly we write 

IG' 0^ 



Gi = 



[0 G"] 



We say that the group G (or Gi) is intransitive, and that 
(yi, y2) and (zi, Z2) form its sets of intransitivity. 

Similarly, if in a group in four variables, these fall into two 
sets of 3 and 1 variables respectively, by a suitable change to 
new variables, the group is intransitive. 

In general, a group G in which the variables fall into two or 
more sets of intransitivity 

[G' 0' 



G = 



G"j 



91] HERMITIAN FORM 207 

either directly or after a suitable choice oj new variables shall 
be said to be intransitive. If such a division is not possible 
we say that G is transitive. 

EXERCISES 

1. The group of order 6 listed in 83 is transitive. It contains an 
intransitive subgroup of order 3 and three intransitive subgroups of order 
2 each. 

2. Find the largest intransitive subgroup contained in the group of 
order 8 listed in 85. 

3. Prove that if a group in four variables appears as intransitive by 
two different changes of variables, 5~ 'G5 and T~ ^GT, such that the sets 
of intransitivity in 5~ 'G5 contain (2, 2) variables, and in T' ^GT contain 
(3, 1) variables, then a change of variables V' ^GV can be foimd such that 
there will appear at least three sets of intransitivity (2, 1, 1). 

Note. Maschke introduced the term transitive as applied to linear 
groups {Mathematische Annalen, Bd. 52 (1899), p. 363). Jordan denoted 
both intransitive and imprimitive groups (106) by decomposable groups 
(Alti delta Rede Accademia delta Scienze fisiche e matematiche, Napoli, 
t. 8 (1879), p. 6). 



Hermitian Invariant, 91-93 

91. Hermitian Form. If the conjugate-imaginary of a 
quantity w be written w, a positive-definite Hermitian form (or 
simply Hermitian form) is an expression such as 



subject to the conditions that it vanishes only if 
^i=a;2= . . . =Xn = 0, and is real and positive for all other 
sets of values assigned to these variables. 

Theorem 4. A positive-definite Hermitian form J in n vari- 
ables may be reduced to the form 

yiyi-\-y2y2-\- . . . -i-ymym 



208 HERMITIAN INVARIANT [Cu. IX 

by a change of variables of the following type: 

y2 = P2\Xl-\- P22X2, 



yi= P\X\ + ps2X2-\- . . . -^pnXt, 

yn= Pn\Xl-\- Pn2X2-\- +P^+ + PnnOCn. 

Proof. Arranging / according to ocn and Xn we have 

J == J n '^(InnXnXn'irXn-^n X \Xn-^n-\ 1 -^ > 

where Xn-\ represents a linear function oi x\, . . . , Xn-u 
The coefficient qnn is real and positive, since it is the value 

of J obtained by putting Xn = l, Xn-i=x-2= =ici = 0. 
Accordingly, 



and we may write 

\ Vg/\ y/qj qnn 

= ynyn-\-Jn-l: Say, 



where 



yn = ^qnnXn H -=, 

vg 



and is therefore a linear function of a^n, . . . , xi. 

The function Jn-i fulfils the conditions of an Hermitian 
form in n 1 variables ocn-i, . . . , Xi, as it is of the required 
tj'pe and is real and positive for any set of values allotted to 
these 1 variables except 0, . . . , 0. For, it is the value 
of Jn obtained by putting Xn= Xn-i/qnn- Hence, we may 
arrange Ja-i according to Xn-i and ic-i and proceed as above. 
We find 

Ju-l =yn-iyn- 1 -\-Jn-2, 



92] CONJUGATE-IMAGINARY GROUPS 209 

where yn-i is a linear function of x-i, x-2, . . . , Xi. Con- 
tinuing thus, we finally prove the theorem. 

92. Conjugate-imaginary Groups and Invariant Hermitian 
Form. If in a group G we replace the variables xi, . . . , Xn 
and the elements a^ of the matrices by their conjugate- 
imaginary values xi, . . . , Xn, a,t, we evidently obtain a group 
G simply isomorphic with G. We shall say that either group 
is the conjugate-imaginary of the other. 

We say that an Hermitian form J is invariant under a group 
G, or that / is an Hermitian invariant of G, when / is trans- 
formed into itself by the (intransitive) group in 2 variables 
xi, . . . y Xn, xi, . . . , Xn made up of G and G. 

Theorem 5. There is always an invariant Hermitian form 
of a given linear group G in n variables * 

Proof. Let the transformations of the group made up of 
G and G be denoted by Ti, T2, . . . , Tg, and let / represent 
the Hermitian form xixi-fa;2a;2+ . . -\-XnXn. Then the sum 

j={i)Ti-\-(i)T2-\- . . . +(/)r 

is an invariant Hermitian form of G. 

First, / is an Hermitian form. For, each of the terms 
{I)Ta is the sum of n expressions {xsXt)Ta = XtXs which are 
real and non-negative. The function / is therefore real and 
non-negative, and cannot vanish unless every term {I)T 
vanishes. But, if Ti represents the identity, {I)T\=I and 
does not vanish unless every variable xi, . . . , Xn vanishes. 
This is therefore also the case with /. 

Second, / is transformed into itself by Ti, r2, . . . , Tg. 
For, evidently 

(7)r=((/)ri)r+ . . . +((/)r,)r=(/)ri+ . . . -f(/)n, 

where 

rfi = TffT^. 

*This theorem was proved for =3 by Picard and Valentlncr (1887, 1889), 
and for any n by Fuchs. Moore and Loewy (189(i). Sec Encyklopddie der Maihe- 
malischen Wissenschaflen, Leipzig, 1898-1904, Bd. I, 1; p. 532. 



210 HERMITIAN INVARIANT (Ch. IX 

But, TiTft, . . . , ToTa3.re the transformations Ti, . . . , T, 
over again in some order. It follows that 

(j)r=(/)ri+ . . . -\-{i)T,=j. 

From Theorems 4 and 5 we get the 

Corollary. Such variables Xi, . . . , Xn may be selected 
for a group G that the function 

I=XiXi-j-X2X2-^ . . . +3cJcm 

is an Hermitian invariant of G. 

93. Linear Transformations in Unitary Form. The variables 
of G being chosen as specified in the previous corollary, let 

-4=[a], ^=[a] 

represent corresponding transformations of G and G. Operat- 
ing upon / by -4 and A, we find the following conditions that 
/ may be reproduced: 

aitau+a2ta2t+ . . . +atat = l (^ = 1, 2, . . . , w), 

aitfiii-{-a2ta2i-\r . -\-aniflni=0 {k,l = l,2, . . .,n;k9^l). 

The transformation A fulfilling these conditions is said to have 
the unitary form, or to be a unitary transformation. 
The inverse of A can here be written down at once: 

A~^ = [a'st] {a'u = au). 

For, the condition A~^A = \ht identity leads to the equations 

(8). 

Now, since A A ^ = the identity also, we obtain the follow- 
ing set of equations as consequences of (8) : 

atiati+at2ak2-\- . . +atnatn = l (^ = 1, 2, ... , ), 

atian-\-at2ai2-\-. . . +atuain = {k,l = l,2,. . . ,n;k9^l). 

94. Reducible and Irreducible Groups. A group G is said 
to be reducible if, by a suitable choice of variables, it can be 
written in the symbolic form (cf . 90) : 

fC 
G = 



94] 



REDUCIBLE AND IRREDUCIBLE GROUPS 



211 



that is, if a certain number of the n variables, say xi, . . . ,Xm, 
where m<n, are transformed into linear functions of them- 
selves by every transformation of G. 

For instance, a group in two variables is reducible if (either directly 
or after a proper change of variables) all the matrices are of type 

'a 0^ 

b c 

If this is not the case, the group is said to be irreducible. 
We shall say that the m variables xi, . . . , Xm form a reduced 
set for G. 

Theorem 6. A reducible group G is intransitive, and a 
reduced set constitutes one of the sets of intransitivity of G. 

Applied to the illustration above, the theorem asserts that the group 
there given can be written in the form 

a o' 

by a suitable choice of variables. 

Proof. The group G has an Hermitian invariant, which 
by the change of variables specified in 91 may be written: 

yiyi-\-y^2-\- . +ynyn. 

Making the corresponding changes in G, this group is still 
seen to be of the form 

IG' 



G" G' 



namely, 

(10) ^*='""'/ + 
yt=(h\y 1+ 



. +<Im/m (5 = 1, 2, ... , fn), 

. \-(hmy'm+ . . . -\-atny'n 

(/ = w4-l, W4-2, ...,). 

Applying the conditions (8) and (9), 93, and writing Cm for 
the product a^a, we obtain, among others, the following 2(n m) 
equations: 

c+i.+ . . . +<:,= 1 (t) = w+l, m+2, . . . . n), 

Ci + . . . +<:. = ! (u' = m+l, m4-2, . . . , n). 



212 CANONICAL FORM fCn. IX. 

If we now subtract from the sum of the last nm equations 
the sum of the first n-m, we get 

n _ m 
*-m+l (-1 

The quantities Cu being real and non-negative, it follows that 
those which enter into this sum all vanish. Moreover, since 
a,t = follows from ^^ = 0, the equations (10) now take the form 

v.=a.iyi4- . . . +asmy'm is = l, 2, . . . ,m), 

yt=atm+iy'm+i-\- . . . +atny'n {t=m+l, w+2, ...,), 

and the theorem is proved. 

EXERCISES 
1. Prove that if 

'p q 



A = 



r s 



is a unitary transformation, then r= q and s=p. 

2. Prove that if all the elements of the matrices of the transformations 
of G are real, then there is a quadratic function of the variables which is 
invariant under G. 

95. Theorem 7. A linear transformation of finite order 
will assume the canonical form ( 80) hy a suitable choice of 
variables. 

Proof. Let the transformation be 

n 

A: Xs=^ ^ astx't (5 = 1, 2, ... , n). 

t-i 

Then we can always find a linear function yi = biXi + . . . -\-bnXn 
which is transformed into a constant (d) times itself by A 
(i.e., yi is a relative invariant oi A). For, we get 

n n n n 

(yi)A = 2L,^' Z^astx,= 2^x, 7^Mi, 
-i 1-1 (-1 -i 

and this expression is 6yi provided the following equations 
are true: 

n 

eb,= ^b,a (/ = 1,'2, . . . , ). 

-i 



S 96: ABELIAN GROUPS 213 

By tJie theory of linear homogeneous equations, a set of 
solutions bi, . . . , bn, not all zero, of these equations, can 
always be found if is a root of the characteristic equation 
of^ (89). 

If we now introduce new variables such that yi is one of 
these, the group generated by A is reducible, since 

(yi)A=eyi, (yi)A^ = ef^yi, etC, 

and therefore 

yi = ey\ 

is one of the equations specifying A in its new form. By 
Theorem 6, the group generated by A is intransitive, one of 
the sets of intransitivity being yi. Let (y2, . . . , yn) form the 
other (temporary) set of intransitivity. 

The above process may now be repeated for the set (y2, 
. . . , yn)- We determine the linear function C2y2 + . . . +cy 
which is a relative invariant of A, and introduce this function 
as one of n 1 new variables to take the place of y2 , . . . , y. 
Continuing thus, the transformation A will finally appear in 
the canonical form. 

96. Theorem 8. In any given ahelian group K ( 26) of 
linear transformations, such new variables may be introduced 
that all the transformations of K will simultaneously have the 
canonical form. 

Proof. If the group contains only similarity-transforma- 
tions, the theorem is self-evident. Hence we assume in K 
a transformation S which is not a similarity-transformation. 
Let the variables of the group be chosen such that 5 appears 
in the canonical form 

5'f=(ai, . . . , ai; a2, . . i , a2; . . . ; a, . . . , a.), 

the variables being arranged so that those having the same 
multipliers are grouped together. Let there be a variables 
x\, . . . , Xa having the multiplier a\\ b variables Xo+i, . , 
Xa+b having the multiplier a2, etc. 

Now let T be any transformation in K. Since TS=STf 



214 



CANONICAL FORM 



[Ch. IX 



we find by applying the rule for forming products that T has 
the form 

Xt = a,\x'\-\- . . . -\-a,ax'a (5 = 1, 2, ... , a), 

Xt = ata+ix'a-^i-\- . . +ata+e^'o+ft (/ = a+l, a+2, . . . ,a+6), 



Hence we infer that K is intransitive, and if we now confine 
our attention to one of the sets of intransitivity, we may apply 
the above process to that set. This will, therefore, break up 
into further sets of intransitivity. Continuing thus, the ulti- 
mate sets of intransitivity contain one variable each, and the 
theorem is proved. 

Instead of the phrase " let the variables be so chosen that a (given) 
transformation (or group) will appear in the canonical form " we shall 
often say simply: " let the (given) transformation (or group) be written 
in canonical form." 



EXERCISES 



1. Can the transformation 



1 



1 1 

be reduced to the canonical form? Find the condition that the trans- 
formation 

a 



which is not necessarily of finite order, can be written in canonical form. 

2. Prove that an abelian group in two variables can be written in 
canonical form and that at the same time the Hermitian invariant becomes 



CHAPTER X 
THE LINEAR GROUPS IN TWO VARIABLES 

97. Introduction. We shall limit ourselves to the deter- 
mination of the groups whose transformations have unity for 
the value of their determinants. From these all other forms 
of groups may readily be constructed (cf. 83-87). We 
shall say that a given group is a type of all groups which may be 
obtained from it by a mere change of variables. 

All the types we encounter (with one exception) contain 
a group of similarity-transformations of order 2, E = {\, 1), 
E\ = { \, 1), due to the fact that a linear transformation 
of determinant unity and of order 2 does not exist unless it 
be the similarity-transformation Ei. A transformation whose 
corresponding collineation is of order 2, if written in canonical 
form, must necessarily be {i, i), where P=l. The excep- 
tion mentioned is the type of an abelian group of odd order. 

Following Jordan, it shall be our practice to call the order of 
a linear group g(t>, if g is the order of the corresponding collinea- 
tion-group, and (f> the order of the subgroup of similarity- trans- 
formations contained in the given group. 

There are several processes available for the determination 
of the types of groups sought.* We shall here employ a modi- 
fied form of Klein's original process, which depends largely on 
geometrical intuition. 

Klein, Mathematische Annden, Bd. 9 (1876), p. 183 ff.; Vorlesungen iiber 
das Ikosaeder, Leipzig, 1884, pp. 11&-120. Goidam, MalhenuUische AnnaJen, Bd. 
12 (1877), p. 23 ff. Jordan, Journal fur die reine und angewandte Mathemaiik, 
Bd. 84 (1878), pp. 93-112; Atti della Reale Academia di Napoli, t. 8 (1879). 
Fuchs, Journal fiir die reine und angewandte Mathematik, Bd. 81, 85 (1876, 1878), 
pp. 97, 1 ff. Valentiner, De endelige Transformaiions-gruppers Theori, Copen- 
hagen, 1889, p. 100 ff. 

216 



216 THE LINEAR GROUPS IN TWO VARIABLES [Ch. X 

98. Outline of the Process. 1. Let G be a group in two 
variables xi, X2- _Then by the introduction of the conjugate- 
imaginary group G (cf. 92) and by the selection of new vari- 
ables X, Y, Z which are bihnear in xi, X2 and their conjugate- 
imaginary values ^1, X2, we obtain a group G' of real rotations 
in space, leaving the origin fixed (99). 

2. Consider now a sphere 2 of radius 1 whose center is the 
origin. With each rotation of G' belongs an axis of rotation. 
One of the points where such an axis pierces 2 together with 
all those points into which this point is moved by G' form the 
vertices of a regular polyhedron, including the limiting cases 
where there is a single axis of rotation or where the polyhedron 
becomes a flat polygon ( 100). 

3. The determination of G' is therefore made to depend 
upon the construction of the analytical expressions represent- 
ing the rotations of the regular solids. We find five different 
types for G' and correspondingly five different types for the 
linear groups G ( 101-103). 

99. The Group of Rotations G'. Let 

a b 



S = 



c d 



be any transformation of G, whose variables xi, X2 are chosen 
such that the Hermitian invariant is/ = a:iici+a;2iC2 (Cor., 92). 
Then the following equations are true (93, 97): 

ad hc = \=ad he, 

aa-j-bb = l, ac-{-bd = 0, cc-\-dd = l. 

From these we obtain 

c=b, d = a, 

Moreover_^ if we let p, q represent the positive square roots of 
aa and bb respectively, and put a = pa, b = q^, we get, since 
P=P> l^r- 

C=-q^, d = pa; 



991 THE GROUP OF ROTATIONS G' 

Furthermore, if we put 7 = VajS and 5 = Va/zS, we have 

77 = 55 = 1. 
Then it follows by direct multiplication ( 78) that 

S=S\S2Sz, 



217 



where 



Sx = 



'7 




P q 




'5 0' 




, S2 = 




, ^3 = 




.0 7. 




-q P. 




.0 "5. 



The corresponding transformation 5 of the conjugate- 
imaginary group G can similarly be written as a product SiS2S'd, 
where 



7 0- 


, 52 = 


P q 


, s,= 


"5 0' 


.0 7. 




-q P. 




.0 5, 



In these expressions we shall finally put 

7 = cos tt f sin , S = coswisinw, p = cosv, 

u, V, w being real angles. 

We now introduce the new variables 



q = smv, 



X = xiit:i a:2a;2, Y =xiX2-\-X2Xi, Z=i{x\X2X2X\) 

These are transformed into linear functions of themselves 
by 5i, 52, 53, operating simultaneously with S\, S2, S3. In 
fact, we find 

(X)SiSi =X, {Y)SiSi = Y cos 2u-Z sin 2u, 

{Z)SiSi = Y sin 2m +Z cos 2u; 

or, to follow our previous practice, 

10 

5i5i= cos2w -sin2tt 

sin 2w cos 2k 



218 



THE LINEAR GROUPS IN TWO VARIABLES (Ch X 



Similarly we find 

cos 2v sin 2v 



S2S2 



sin 2^ cos 2v 











5353 = 



COS 2w sin 2w 



sin 2w cos 2w 



If we interpret X, Y, Z as rectangular coordinates in ordi- 
nary space, we recognize here three real rotations around the 
X-, Z-, X-axes respectively, the origin remaining fixed. The 
rotations performed successively will, as is well known, be 
equivalent to a single rotation. With the transformations 
of the group G are therefore associated rotations which evi- 
dently form a group G' isomorphic with G. The isomorphism 
is (1, 2) in the case where G contains i = ( 1, 1); other- 
wise it is (1, 1), since we may readily prove that to identity 
of G' will correspond only E = {1, 1) or Ei of G. In other words, 
G' is simply isomorphic with the collineation-group correspond- 
ing to G. 

100. The Regular Polyhedron. Consider an axis of rota- 
tion (L) of G', and let the various angles of rotations around 
L be the different multiples of (360/w); we shall say that 
L is of index m. Let Pi be one of the points where L cuts the 
sphere 2. This point will be transformed into (say) k distinct 
points upon 2 by G': Pi, P2, . . . , Pt, all of which will be 
extremities of axes of rotation of index m. The distribution 
of these points about any one of them is similar to the distri- 
bution about any other. 

Now let arcs of great circles be drawn connecting Pi with 
all the other points P2, . . . , Pt, and let the shortest arc be 
of length A. The number of arcs of this length radiating 
from Pi is w or a multiple of m, since always m of the arcs are 
interchanged by rotations about L through the different multi- 
ples of (360/ w). However, there cannot be more than 5 arcs 
A ; an exception occurring where we have just one or two 
points Pi, P2, one or both extremities of L, in which case A = 360 
or 180. For, if there were 6 or more, a pair of them (say 
C, C^ would make an angle /3^60 with each other at Pi; 



lOOl THE REGULAR POLYHEDRON 219 

and, this being the case, the arc B connecting P, and P, (the 
points of Pi, . . . , Pt located on C, and d) would have a 
length <A. For, by trigonometry, 

cos B = cos^ A +sin2 A cos /3 ^ cos^ A -\-\ sin^ i4 > cos -4 ; 

and, since 0<5<90, it follows that B<A. But this is con- 
trary to hypotheses, since the lengths of the arcs radiating 
from P., are equal to the lengths of the arcs radiating from Pi. 

Let m>2. Then it follows that there are just m arcs of 
length A radiating from Pi, each making an angle of (360/m) 
with its adjacent arcs. The same will be true for each of the 
points P2, . . . , Pt, and we see readily that the sphere will 
be divided by all the arcs of length A , joining the various points 
Pi, . . . , Pt which can be reached from one of them by pass- 
ing along such arcs, into a number of equal and regular polygons. 
Accordingly, these points, say Pi, . . . , Pi, are the vertices 
of a regular polyhedron inscribed in S. 

Consider next the case where there are no axes of index 
greater than 2. Proceeding as above, we let L denote an axis 
of index 2, and we obtain the points Pi, . . . , Pt by G'. There 
are at least two arcs of length A radiating from Pi making an 
angle of 180 with each other. Taken together they form a 
single arc C upon which (when extended round the sphere) 
Pi and some other points P2, . . . , Pj lie, equally distributed 
over the entire circle. If / > 2, a rotation of 180 around Pi 
followed by a rotation of 180 around one of the points next 
to Pi is equivalent to a rotation of (720//) around an axis 
perpendicular to the plane of the circle C 

Every axis is of index 2 by assumption. It follows that 
2// = l or 1/2; i.e., / = 4, In this case we have three mutu- 
ally perpendicular axes of index 2. 

The distance A is therefore either 180 or 90. In the for 
mer case we have a single axis of index 2 in G'. In the latter 
case there are four arcs of length A radiating from Pi, lying 
on two circles which are at right angles to each other at Pi. 
Their extremities lie in the diametral plane which is perpen- 
dicular to the axis L, and must be 90 or 180 apart. Con- 



220 THE LINEAK GROUPS IN TWO VARIABLES [Ch. X 

sequently, G' contains just one axis of index 2, or just three such 
which are mutually perpendicular. 

The Groups of the Regular Polyhedra, 101-103 

101. Limiting Cases. We notice first that the most general 
linear homogeneous change of variables {xi, X2) in G is indicated 
by a linear transformation T ( 88) to which again corresponds 
the most general rotation of 2 about its center. It follows that 
any given configuration arrived at in 100 may at the outset 
be placed in any required position relative to the axes of 
coordinates X, Y, Z. 

Beginning then with the simplest case where there is a single 
axis L of rotation, we let this be the X-axis. Then sin2z) = 
and cos22; = l (cf. 99). Hence S has the form (a, dba"^). 
If S is of order g we have (0:)'' = 1. 

(A) G': a single axis of index g; 

G: an abelian group (intransitive) of order g: 

5x = (e\ 6-^); X=l, 2, . . . , g; e'' = l. 

The next case to be considered is where there is an axis L of 
index g, assumed to be the X-axis as above, in addition to g axes 
of index 2 lying in a plane perpendicular to L. Let one of the 
latter be the Z-axis; we then have cos 2z;= 1, cos 2{uw) = 1, 
and the corresponding transformation of G is found to be 



T= 



1 
=F1 



(B) Dihedral Group. 

G' : one axis L of index g and g axes of index 2; 
G: 'an imprimitive group of order 2^0 consist- 
ing of the transformations 



5x=fdze\ e-^), n= 




102) 



THE TETRAHEDRON AND OCTAHEDRON 



221 



102. The Tetrahedron and Octahedron. We now examine 

the five ordinary regular solids. Of these, the hexahedron and 
octahedron furnish the same set of axes of rotation, as do also 
the dodecahedron and icosahedron. We therefore have only 
three cases to consider: the tetrahedron, octahedron and icosa- 
hedron. 

In the case of the tetrahedron we have four vertices and 
correspondingly four axes of rotation of index 3; besides, three 
axes of index 2, each passing through the middle points of a 
pair of opposite edges. The latter axes are mutually perpen- 
dicular and may be taken as the X-, Y-, and Z-axes. The 
corresponding transformations of G are then as follows: 



Ti = ii, -i), T2 = 



'0 i 


, T,= 


' 1 


[i OJ 




[-1 oJ 



either directly or after multiplication by jEi = ( 1, 1). If 
the vertices are named a, b, c, d, the three rotations permute 
them among themselves according to the substitutions 

(ab){cd), (ad) (be), (ac){bd). 

The remaining rotations permute the vertices three at a time 
cyclically, as (abc), . . . The corresponding transformations 
of G may be determined analytically from the conditions that 
they are each of order 3 and transform the coUineations corre- 
sponding to Ti, T2, T3 cyclically. Certain ambiguities arise 
from the fact that the similarity-transformation JSi = ( 1, 
1) is present in the group. Thus, S = (abc) may transform 
Ti into T2 or into T2Ei, etc. For (abc) we find four forms 
possible, all of which are present in G if one of them is. We 
shall choose the following form : 



-l+i -!+ 



S = 



2 2 

1-f-t -l-i 



222 THE LINEAR GROUPS IN TWO VARIABLES [Ch. X 

(C) Tetrahedral Group. 

C : generated by 5 = (abc) and Ti = (ab) (cd) ; 
G: a primitive group ( 106) of order 12<^ gen- 
erated by the transformations S and Ti above. 

The rotations of the octahedron include those of the tetra- 
hedron S = {abc), Ti = {ab){cd) if here a, b, c, d represent each 
a pair of opposite faces. To the list of generating rotations 
we now add one, U say, of order 4, having the same axis as 
T\, and lP = Ti, or U = (acbd). The corresponding trans- 
formation of G is readily found to be 



VV2 V2/ 



(D) Octahedral Group. 

G' : generated by 5 = (abc) and U= (acbd) ; 
G: a primitive group of order 24<f>, generated by 
5 and U above. 

103. The Icosahedron. An icosahedron contains 10 axes 
of rotation of index 3. These (counted twice) may be grouped 
in 5 sets of 4 axes each, such that the axes of each set are 
arranged in the same way as the axes of index 3 in (C). In 
this manner we obtain 5 regular tetrahedrons, each correspond- 
ing to a subgroup of order 12. The group is accordingly iso- 
morphic with the alternating group of order 60 and is generated 
by the substitutions S, Ti of (C) and a substitution V = {ab){de), 
whose corresponding transformation may be determined from 
the relations 

F2= or El, {TiVf = E or Ei, {SVy=E or i. 

The first and last ambiguities fall away, as of necessity 
V^ = {SV)^=Ei (cf. 97); and by using VEi if necessary in 
place of V we may take (TiV)^=E. We then find 

i ^-'^ 

v= 



103] 

where 



THE ICOSAHEDRON 223 



^ I-V5 I+V5 

(E) Icosahedral Group. 

G': generated by S = {abc), T\ = {ah){cd) and 

V^{ah){de)\ 
G: a primitive group of order 600 generated 
by S, Ti and V above. 

EXERCISES 

1. Construct the analytical forms of the rotations of G' corresponding 
to the generating transformations Ti, Ti, Tz, S, U and V. Prove that the 
X-, Y-, Z-axes are permuted among themselves by all of these rotations 
except V, and hence that the group G' in the cases (C), (D), as a group in 
three variables, is imprimitive (cf. 106). 

2. Determine a set of generators of (E) corresponding to the substi- 
tutions S'= {abode), U'={ad){bc), T'={ab){cd) of the alternating group 
in five letters; under the condition that S' is written in canonical form: 
5'=(*',.*);*'=1. 

Hints: We first determine U' from the condition 1}'-^S'U'=S'-^: 



U'= 



p 



Pq=-l. 



The change of variables x=pX, y=Y will leave S' unaltered and will 
reduce U' to the form above, except that now P=l, q= l. We finally 
assume 

'a /3" 



r= 



y 5 



aS-0y=l. 



The condition T'^=Ei (97) gives us a + S = (t)+(-t)=0 (89), which, 
combined with f/'-^rf7'=r or r, (only T'E, will be compatible with 
the previous results) is equivalent to 5= a, 7 = 0; a*+/3*= 1. Finally, 
we have the relation (abcde)-(ab)(cd) = (bde); that is, the transformation 
S'T' is of order 3 and its characteristic roots are consequently w, w* or 
w, w*. The latter possibility can be avoided by taking T'Ei instead of 
T'. Applying this condition we finally have 



""Vf' ^"Vfi- 



224 THE LINEAR GROUPS IN TWO VARIABLES (Ch. X 

Invariants of the- Linear Groups in Two Variables, 

104-105 

104. General Theory. A homogeneous function of the vari- 
ables xi, X2 of a group 

G: Si, 02, , Sff^ 

is called an invariant of G (or we say that G leaves f invariant) 
when/ is transformed into a constant multiple of itself by G: 

(f)Sj = c^. 

Let / be resolved into linear factors. These are permuted 
among themselves by G, and the product of a set of them which 
are permuted transitively will evidently furnish an invariant 
by itself. This invariant, say F=fif2 ...//,, can readily 
be constructed by operating upon one of the factors /i by the 
transformations of G, and we shall call it a fundamental invari- 
ant. Any invariant is accordingly a product of fundamental 
invariants. 

If /i be selected at random, the corresponding fimdamental 
invariant is evidently of degree g. To obtain fundamental 
invariants of lower degree we make use of a theorem of transi- 
tive substitution groups, namely that the ratio g<f)/h is the order 
of that subgroup of G which leaves /i invariant. 

Now this subgroup, Gi, must be abelian. For we may 
change the variables, introducing /i as one of the new variables, 
say xi. Then Gi must appear in the form of a reducible group: 

a 0' 

and can accordingly be written as an intransitive group 

a 0' 

But this is the canonical form of an abelian group. It follows 
furthermore that two subgroups, d and G2, having in common 
a linear invariant /i, generate an abelian group. 



105] THE FUNDAMENTAL INVARIANTS 225 

The other factors /2, . . , /a of F are linear invariants 
of subgroups G2, . , Gh of G, conjugate to Gi. For, if 
Sa~^GiSa = G2, then (fi)Sa=f2 belongs to F and is an invariant 
of G2. Hence our problem becomes one of determining the 
different conjugate sets in G of abelian subgroups which are 
not subgroups in larger abelian subgroups. The fundamental 
invariant F will be made up of one factor for each of the 
subgroups of the set if there is no transformation in G which 
transforms one of the linear invariants of Gi into the other. 
In other words, if Gi be written in the canonical form (a, c), 
and if there is no transformation in G of type 

/>' 

[q 0. 

we get two fundamental invariants by starting with /i = Xi 
and/i =X2; otherwise we get just one invariant, containing both 
xi and X2 as factors. 

105. List of the Fundamental Invariants. We shall, of 
course, hmit ourselves to the invariants of degree <g. 

(A) Case (A), 101. Two invariants, Xi and X2. 

(B) Dihedral Group. One invariant, iCiX2. 

(C) Tetrahedral Group. There are two conjugate sets of sub- 
groups, of orders 20 and 30. The first set consists of groups 
conjugate to that generated by Ti, and the invariant is * 

t=XiX2(Xi* X2'^). 

The second set contains the groups conjugate to that gen- 
erated by S. Here we have two invariants, each containing 
just one of the linear invariants of S: 

^=Xl^ -\-2V^Xi^X2^^-X2*; ^ = OJi" - 2 V^iCi%2Ha:2*. 

These invariants satisfy the relation 

12V^/2_,j>3_|_^3^0. 

(D) Octahedral Group. Here we have three conjugate 
sets of subgroups, of orders 40, 30 and 20. In the first set 

The notation is that given by Klein, Vorlesungm, etc., pp. 51^58. 



226 THE LINEAR GROUPS IN TWO VARIABLES [Ch. X 

there is a group generated by C/, and the corresponding invari- 
ant is / above. The second set contains the group generated 
by S, and the corresponding invariant is the product of * and 
tjf above ' 

The third set contains the group generated by UT2, and we 
get the invariant 

X = xi^^ - SSxi^xz"^ - ZSXi^Xz^ -{-X2^. 

These invariants satisfy the relation 

(E) Icosahedral Group. We shall take the group as repre- 
sented in Exercise 2, 103. There are three sets of subgroups 
of orders 20, 30 and 50, containing the groups generated by 
/', S'T' and S' respectively. We get, correspondingly, the 
three invariants 

T=Xi^^-\-X2^^+h22{xi^H2^-Xi^X2^^) - 10005(:i20a:2^o+xii0x220), 

E= -Xi^^-X2^^ + 22%{xi^H2^-XiH2^^)-AQ^Xi^^X2^^, 
J=X\X2{x\^^-\-\\Xi^X2^ X2^^), 

which satisfy the relation 

r2+^-1728/s = 0. 

EXERCISES 

1. The invariant * of the tetrahedral group is the Hessian covariant 
(cf. 174) of the function *: 

48V33*= *" *" 
and the invariant / is the Jacobian of the functions * and *; 

4>i 4>2 



-32V^/= 



^2 



Obtain similar relations for the octahedral and icosahedral groups. 

2, From the fact that no two abelian subgroups of G can have a trans- 
formation in common (except similarity-transformations) unless they 
generate a larger abelian group, it follows that G is made up of a number 
of distinct abelian groups ^1, Ht, . . . , having no transformations in 



105] THE FUNDAMENTAL INVARIANTS 227 

common except E and Ei. Hence, if the orders of these groups be re- 
spectively Ai, . . . , we must have 

(1) g<t>=2*+(hi4>-2*)+(h24>-2)+ . . . 

Now, hi, . . . are factors of g, say hi=g/gi, . . . , and there are either 
gi/2 or gi subgroups conjugate to Hi, according as there is or is not a trans- 
formation in G which permutes the linear invariants of Hi. Hence, adding 
the corresponding terms in the right-hand member of (1), we obtain 

g0= 2 -I- ^ igi(A,-2) -f ^g'i(A',<^-2) 



or 



^=\-x^HhxH} 



Verify this (Diophantine) equation for the groups (A) to (E). 

3. Prove that, in the case of (D) or (E) , any invariant of degree g, say 
/, is an absolute invariant; that is, it is transformed into itself by every 
transformation of G. 

Prove also that / is a rational integral function of two of the three 
fundamental invariants listed above for the respective group. 

* Counting the transformations E and Ei once each. 



CHAPTER XI 

SOME SPECIAL TYPES OF GROUPS 

106. Primitive and Imprimitive Groups. Let us suppose 
that the group G, 90, contains not only transformations 
of type A , but also some of type 

p q t V 

q p V t 

u w r s 



5 = 



w u 



s r 



which upon the change of variables there employed becomes 
p-q t-v 

uw rs 

p-\-q t-\-v 



B = 



u-k-w r+5 











fO B' 
B" 



then we say that G is imprimitive, under the assumption that it 
is transitive. 

In general, a transitive group G, in which the variables (either 
directly or after a suitable choice of new variables) can be separated 
into two or more sets Yi, . . . , Yt, such that the variables of each 
set are transformed into linear functions of the variables of the sawe 
set or into linear functions of the variables of a different set, is said 
to be imprimitive. If such a division is not possible, the group 
is primitive. The sets Fi, . . . , F* are called sets of imprimi- 
tivity, 

228 



107) IMPRIMITIVE GROUPS 229 

107. Theorem 9. Let G be an imprimitive linear group in 
n variables. These may be chosen in such a manner that they 
break up into a. certain number of sets of imprimitivity Fi, . . . , 
Yt of m variables each {n = km), permuted according to a transi- 
tive substitution group K on k letters, isomorphic with G. That 
subgroup of G which corresponds to the subgroup of K leaving one 
letter unaltered, say Y\, is primitive as far as the m variables of 
the set Yi are concerned. 

If m = \, k = n, then G is said to have the monomial form 
or to be a monomial group. 

Proof. Let the variables of G break up into say k' sets Fi, 
. . . , Fjf, permuted among themselves according to a sub- 
stitution group K' on k' letters. This group K' is transitive 
(as a substitution group, 12); otherwise G would not be a 
transitive linear group. Hence i^'. contains k' l substitutions 
S2, S3, . . . , Sf which replace Fi by F2, F3, . . . , Yt' 
respectively. We shall select k' l corresponding transforma- 
tions of G and denote them by A2, A3, . . . , Af. The condi- 
tion that the determinants of these transformations do not 
vanish, implies that the sets contain the same number of 
variables n/k'. 

There is in K' a subgroup K\ whose substitutions leave 
Fi unaltered ( 12). This subgroup, together with the sub- 
stitutions S2, . . . , St' will generate K'. Correspondingly, 
G is generated hy A2, . . . , At> and that subgroup G\ of G 
corresponding to K' \, and which therefore replaces the variables 
of Y\ (say yi, >'2, . . . , >) by linear functions of the same 
variables. If we now fix our attention upon just that portion 
of each transformation of G which affects only these m variables 
and which plainly forms the transformations of a linear group 
[Gi] in m variables, we shall prove that if [d] is not primitive, 
then new variables may be introduced into G such that tlie number 
of new sets of imprimitivity is greater than k' . 

Accordingly, let the variables of [d] break up into at least 

two subsets of intransitivity or imprimitivity, say Fi"\ . . . , 

Y\:^ . New variables will now be introduced into the sets F2, 

. . . , Ffc* such that At will replace Fi"\ . . . , Y ^^ by dis- 



230 SOME SPECIAL TYPES OF GROUPS [Ch. XI 

tinct subsets F/", . . . , F/*^. In this manner the variables 
of G will be divided into Ik' subsets, and it remains for us to 
prove that any transformation 5 of G will permute these sub- 
sets among themselves; that is, 5 will transform the variables 
from any one subset into linear functions of the variables 
of one of these subsets. 

Let 5 replace F^, by F^. Then AaSAfi~^ = T transforms 
Fi into itself; that is, T is a transformation of Gi and will 
therefore permute among themselves the subsets F/", . . . , 
Fi^". It follows that the transformation Aa~^TA0 = S will 
transform any subset of Ya into some subset of Yp, and the 
proposition is proved. 

We can therefore keep on changing the variables so as 
to increase the number of sets of imprimitivity, until the sets 
contain just one variable each, or until the group [Gi] is primi- 
tive. The theorem is therefore proved. 

108. Lemma. A linear group G having an invariant abelian 
subgroup H whose transformations are not all similarity-trans- 
formations is either intransitive or imprimitive. 

Proof. Write H in canonical form. The variables can then 
be arranged into sets having the property that a transforma- 
tion of H affects all the variables of any one set by the same 
constant factor. 

To illustrate, let H be generated by the transformations 

Ti={fxi, ai, ai, ai, 02) (ais^aj), 

Ti=(pi, Pi, Pi, P2, Pi) (/3i7^/3i)' 

Here we have three sets: X= (xi, 0:2), Y= (xi, Xt), Z= (xt). 

Then it is readily proved that G permutes these sets among 
themselves. Thus, in the illustration given, let 5 be a trans- 
formation of G and r<, Tj of H, and let S-^TiS = Tj. Now 
suppose that the variables of X are transformed by S into 
two variables yi, y2 forming a set X'; we must then prove that 
X' is either X or F. We have 

(xi, X2)TiS = (xu X2)STj, 
or 

Oiiyi, y2)=(yi, y2)T,. 



108] SYLOW GROUPS 231 

Hence, the variables yi, y2 are transformed by Tj into the 
same multiple of themselves. Since Tf may be taken to repre- 
sent any transformation of H, it follows that yi, y2 are linear 
functions of xi, X2, or linear functions of X3, x^. 

Theorem 10. A linear group whose order is the power of a 
prime number can be written as a monomial group by a suitable 
choice of variables xi, . . . , Xn] that is, its transformations 
have the form: * 

Xs = astx't (s = l, 2, . . . , ; t = l, 2, . . . , n). 

Proof. 1. A group P whose order is the power of a prime 
number p is either abelian or it contains an invariant abelian 
subgroup Q whose transformations are not separately invariant 
in P ( 48). In the first case the theorem follows from Theorem 
8, 96. In the second case, Q can be written in canonical form, 
and P is intransitive or imprimitive by the above lemma. If 
the theorem is true for a transitive group, it is evidently true 
for an intransitive group; hence we need merely discuss the 
case where P is imprimitive. 

2. By Theorem 9, P is monomial unless there is a group 
[Pi] which is primitive in the m variables of a set Fi, But the 
order of [Pi] is again a power of p, and this group cannot there- 
fore be primitive, by 1. 

Corollary. A linear group in n variables whose order is 
the power of a prime greater than n is abelian. 

109. Theorem 11. A linear group G in n variables and of 
order g = g'p^q^r^ > where p, q, r, . . . are diferent primes 
all greater than n-\-\, contains an abelian subgroup of order 
p"q^r<^ . . . 

To prove this theorem by the process of complete induction, 

we assume it true for any group whose order is divisible by a 

factor of pf^q^r^ , smaller than this number. We shall 

also assume the theorem true for a transitive group in fewer 

than n variables; it will then immediately be true for an 

intransitive group in n variables. 

* Proofs of this theorem were given by the author in Transactions of the 
American Mathematical Society, vol. 5 (1904), pp. 313-314; vol, 6 (1905), p. 232; 
and by Burnside, Theory f Groups, second edition, Cambridge, 1911, p. 352. 



232 SOME SPECIAL TYPES OF GROUPS [Ch. XI 

We therefore assume that G is transitive. At the outset 
we anticipate a theorem given below (Cor. 3, 135), from which 
it follows that G contains a transformation of order pqr. . . . 
Now, among all abelian subgroups of G whose orders are of 
the form p^q^r^. . . . , where a^a^l, 6^/3^1, etc., let // be 
one whose order is the highest possible. We shall then prove 
that here 
(1) a = a, ^ = 1, 7=c, . . . 

For this purpose, let us assume that these equations are 
not all true; say a<a. Then H contains a Sylow subgroup P\ 
of order p" which is contained in a Sylow subgroup P of G of 
order p^ (11). The groups H and P being abelian (Cor., 
108), all the transformations of P\ are invariant in both, 
and will therefore also be invariant in the group K generated by 
H and P. But, if such an invariant transformation is not a 
similarity-transformation, K will be intransitive (cf. proof of 
Theorem 8, 96) and will (by assumption) in this case con- 
tain an abelian subgroup of order p^q^r^ , contrarv to the 
supposition made in regard to H. Again, if the transformations 
belonging to P and invariant in K, say p' in number, are all 
similarity-transformations, the group K contains an invariant 
subgroup (Theorem 12, Ex. 2), the order of which is divisible 
by />""*. Hence, by assumption, it contains an abeb'an sub- 
group of order p''~'q^r'^ . . , The corresponding subgroup of 
K is evidently also abelian and has for its order p^-p'^~*(fr'^ . . . 
= p<^(ff . . . Hence finally, if a < a, i7 cannot be that abelian 
subgroup of highest order p"^r'' . . . contained in G. The 
equations (1) follow. 

EXERCISES 

1. Prove that if n+1 is a prime, and if the order of G is g-g'in+lYp"^ 
. . . , where i> 1, and p, q, . . . are primes greater than n+1, then there 
is in G an abeh'an subgroup of order (n-t-l)VV 

2. It follows from Theorem 11 that a group in n variables whose order 
contains no prime factors smaller than n-\-2 is abelian. Prove that if the 
order contains no prime factors smaller than + l, the group is abelian. 

3. Construct all the types of (monomial) groups of order 2* in three 
variables, and show that such groups contain cither a transformation of 
order 8 or one of order 4 and type ( 1, i, j). 



110] THEOREM ON SIMILARITY-TRANSFORMATIONS 233 

4. Construct all the types of groups of order 3V(0= 1 or 3; cf. ( 111) 
and show that such groups contain a transformation of order 9<f>. 

110. On the Group of Similarity-transformations. The 
necessity for the presence of similarity-transformations in a given 
linear group can be determined from the following 

Theorem 12. // a Sylow subgroup P of order p^ in a linear 
group G of order g can be generated by a group P' of order p^~^ 
and a similarity-transformation T of order p", then there is in G 
an invariant subgroup G' of order g/p which does not contain T. 

Proof. Let H represent the regular substitution group on 
g letters simply isomorphic with G ( 27). Then, if among 
the letters of H, yi, y2, . . . , )* form a transitive set for P', 
the sum 

J=yi-\-y2-\- . . . +yk 

will be transformed into itself by every substitution of P'. 

Let furthermore be a primitive root of the equation ^ 1 = 
(116); then the function 

/=/-i-rK/)r+r2(/)r2-f . . . -\-d-^+HJ)'P'-^ 

is transformed into a constant (d) times itself by T: 

iI)T=^{J)T-\-d-'{J)T^-\- . . . -\-d-'>+'{J)T^ 

= el+d[{J)v-j\ = dl, 

since 7^ belongs to P' and therefore transforms / into itself. 
Moreover, any substitution of P' transforms I into itself. 

For, 

p-i p-i p-i 

{i)s =^6-^)^8 =^e-V)ST=^d-V)'r, 

r=0 r = r-0 

since T is commutative with S. It follows that / is an invari- 
ant of P (the function / cannot vanish identically, since no 
two of the terms J, {J)T, {J)T^, . . . can have a letter in 
common) . 

Again, / is not an invariant of any substitution in G other 
than those in P. For, if i? be a substitution such that {I)R-cI, 
the letters occurring in I must be permuted among themselves 
by R. Let us suppose that R changes yi into >'2. But, the 



234 SOME SPECIAL TYPES OF GROUPS (Ch. XI 

letters of / form a transitive set for P; accordingly, there 
is a substitution in P, say ^i, which also changes y\ into y2. 
Then RSi~^ leaves yi fixed and must therefore be the identity; 
that is, R=Si. 

It follows that the substitutions of G transform I into 
just g/p*" functions I, Ii, I2, . > . , no one of which is a con- 
stant multiple of another. The product K = IIil2 ... of 
these functions is therefore an invariant of G. 

Now, all the substitutions of G for which iT is an absolute 
invariant, that is, for which {K)R = K, must form an invariant 
subgroup G' of G, as is easily seen. To this group the substi- 
tution T does not belong, since 

{K)T = ^K, . 
where k = g/p^. For, let If = {I)R, then 

{Ir)T = {I)RT = {I)TR = d{I)R = elr. 

But, ^9^1 since k is prime to p. 

This subgroup G' is of index p. For, the constant mul- 
tipliers of K that result by operating upon K by the various 
substitutions of G are integral functions of 6 and must be 
roots of unity. Such roots can therefore, by 116, 6, be no 
others than powers of 6. Moreover, it is readily seen that 
each power must occur equally often, so that the power 1 
occurs g/p times. Hence the theorem. 

EXERCISES 

1. Prove that a linear group in 3 variables of order 9g in which there is 
no transformation of order 94> must contain similarity-transformations. 

2. Prove that a linear group in n variables which contains a subgroup 
P of similarity-transformations of order />' (/> a prime >), contains an 
invariant subgroup of index />*, to which F does not belong. 



CHAPTER Xn 



THE LINEAR GROUPS IN THREE VARIABLES 



111. Introduction. As in the case of the binary groups 
we shall limit ourselves to the discussion of groups of trans- 
formations of determinant unity, and shall generally write 
g<l> for the order of a linear group whose corresponding collinea- 
tion group is of order g. Moreover, the order of a transforma- 
tion 5 will often be written in the form g4>, when the order of 
the group generated by S is g0. For instance, the order of the 
transformation 

S = (a, a, aw^) (a^ = w; w^ = l) 

may be written either 9 or 3<^. 

Though the orders as written may thus virtually refer to 
collineation groups, it must be kept in mind that all purely 
descriptive terms refer to linear groups. For instance, the 
group of order 90 generated by 



5i = (l, W, W^), S2 = 



would be described as a non-abelian group, though the corre- 
sponding collineation-group is abelian. 

The determination of the linear groups in three variables 
is based upon the following classification: 

1. Intransitive and imprimitive groups. 

2. Primitive groups having invariant imprimitive sub- 
groups. 

3. Primitive groups whose corresponding collineation groups 
are simple. 

4. Primitive groups having invariant primitive subgroups. 

235 






1 











1 


,1 









236 THE LINEAR GROUPS IN THREE VARIABLES (Ch. Xil 

The method for the first two and last classes is more or 
less evident. In the discussion below on the groups in class 
3 some theorems are developed that with slight modifications 
can be extended to groups in n variables, and such generali- 
zations are given in 126, together with a r6sume of results 
on the order of the primitive groups in n variables. An intro- 
duction to the theory of the invariants of the ternary linear 
groups is given in 125. 

112. Intransitive and Imprimitive Groups. We have two 
t>'pes of intransitive groups: 

(A) X\=ax'\, a:2=/3a;'2, xz=^yx'z (abelian type). 

(B) xi=ax'\, X2 = ax'2+hx'-i, Xz=cx'2-\-dx'-i. 

In (B) the variables X2, x-s are transformed by a linear 
group in two variables (cf. Chapter X). 

The imprimitive groups are all monomial. There are two 
types: 

(C) A group generated by an abelian group 

H: xi=ax'i, ii[;2 = /3^'2, x^ = yx'z 

and a transformation which permutes the variables in the 
order (a;iX2a;3). By a suitable choice of variables this trans- 
formation can be thrown into the form 

Ti a;i=a;'2, X2 = x':i, xz=x'\. 

(D) A group generated by //, T of (C) and the transforma- 
tion 

R: xi=ax\, X2 = bx'3- X3'=cx'2- 

113. Remarks on the Invariants of the Groups (C) and (D). 
Interpreting xi, X2, xs as homogeneous coordinates of the plane, 
the triangle whose sides are xi =0, a;2 = 0, r*;3 = is transformed 
into itself by the operators of (C) and (D); in other words. 
xia;2X3 is an invariant of these groups. 

It will later be imperative for us to know under what con- 
ditions there are other invariant triangles. Assuming the 
existence of one such, say 

(1) {aiXi-^a2X2-\-a3X3,){biXi-\-b2X2-\-b3X3){CiXi-{-C2X2+C3X3) =0, 



114-1151 PRIMITIVE COMPOSITE GROUPS 237 

we operate successively by the transformations of H and by 
T. Examining the various possibilities we find that (1) could 
not be distinct from x\X2Xii = unless H is the particular group 
generated by the transformations 

5i = (l, CO, op), 52 = (a), w, w) (0,3 = 1). 

There are then four invariant triangles for (C), namely; 
. - XiX2X2 = 0; 

{Xi -\-X2-\- BXz) {X\ + (jiX2 + (JpOXz) {Xi + (jPx2 + OiBXz) = 

(0 = 1, CO or co^). 

The same triangles are invariants of (D) if this is generated 
by (C) in the form just given and the following special form of i?: 

Xl = X'\, X2=x'3, X-i=x'2' 

114. Groups Having Invariant Intransitive Subgroups. All 

such groups are intransitive or imprimitive. This follows from 
the fact that the type (B) has a single linear invariant x\, 
which is therefore also an invariant of a group containing (B) 
invariantly; * and the fact that a group containing (A) invari- 
antly cannot be primitive by the lemma, 108. 

115. Primitive Groups Having Invariant Imprimitive Sub- 
groups. It was shown above that the types (C) and (D) 
possess either one or a set of four invariant triangles. If they 
possess only one such triangle, a group containing one of these 
types invariantly would of necessity also leave invariant that 
triangle, as may be easily proved. That such a group may 
be primitive, it is therefore necessary that (C) and (D) possess 
the four invariants (2) . 

Let us therefore assume a group G permuting among them- 
selves the triangles (2), which we shall for brevity denote 
respectively hy h, t2, h, h'm. the order as they are listed in (2). 

* Let V be any transformations of a group containing (B) invariantly, and 
T any transformation of (B). Then VTV-^=Ti belongs to (B), and if we put 
{xi)Ti=cau {xi)V=y, we have 

iy)T={y)V-'T,V=ay, 

so that y is an invariant of Ti But, Xi being the only linear invariant for (B), 
it follows that y= cxi. 



238 THE LINEAR GROUPS IN THREE VARIABLES [Ch. XII 

We now associate with each transformation of G a substitu- 
tion on the letters ii, tz, ts, U, indicating the manner th which 
the transformation permutes the corresponding triangles. 
We thus obtain a substitution group K on four letters to which 
G is multiply isomorphic, and the invariant subgroup (C) or 
(D) corresponds to identity of K. No one of the four letters 
could be left unchanged by every substitution of K; for, the 
corresponding triangle would be an invariant of G, and this 
group would not be primitive. Moreover, no transformation 
can interchange two of the triangles and leave the other two 
fixed, as may be verified directly. 

Under these conditions the following possible forms for K 
are found: 

(E') 1, {ht2){hk);* 

(FO 1, {ht2){hk), {hk){t2h), {hh){t2k)\ 

(GO the alternating group on four letters, generated by 
(/ife)(fe/4) and {t2kt^. 

Now, to construct the corresponding transformations we 
observe that the group (D) as given in 113 contains all the 
transformations which leave invariant each of the four tri- 
angles. Furthermore, we note that if a given transformation 
V permutes the triangles in a certain manner, then any trans- 
formation which permutes them in the same manner can be 
written in the form V = XV, X being a transformation of (D). 
For, F'F"^ must leave fixed each triangle, and is therefore a 
transformation X as defined. 

We are now in a position to construct the required groups. 
By direct application we verify that the transformations U, 
V, UVU-^: 

U: Xi=x\, :;2 = a;'2, X3=(>)x'3 {^ = (^); 

(3) V: xi = p(x\-\-x'2+x'3), X2 = p{x\-\-wx'2-\-<^x'3), 

X3 = p(x\-^<a^x'2-\-o}x'3) (p= -); 

\ (i) (a^/ 

UVU-^: xi = p{x'i-^x'2-\-(^x'3), X2 = p(x\-{-(ax'2+0}X'3), 

X3 = p{ux\ +X'2 + UX's) 

* The three different subgroups of order 2 oi K would furnish only one type 
for C, since the three different groups obtained are transformable one into the 
other by a t hangc of variables. 



116] ON ROOTS OF UNITY 239 

permute the triangles in the following manner: 

{t2hh) , (/1/2) (^3/4) , {hh) {t2h) . 

Accordingly, since all the required groups contain a trans- 
formation corresponding to {ht2){hU), every such group must 
contain a transformation XV, X belonging to (D). Hence, 
if G contains (D) as a subgroup, it also contains V. If, however, 

(C) were a subgroup of G, but not (D), then either V is con- 
tained in G, or else XV, where X belongs to (D) but not to 
(C). In this event X may be written XiR, where X\ belongs 
to (C). Hence finally, either V ov RV belongs to G. However, 
V^ = {RVy = R. Thus R, and therefore also V, are contained 
in G in any case. 

Again, if G contains a transformation corresponding to 
{t2ht^ or {tih{t2h), such a transformation can be written XU 
or XUVU~^, X belonging to (D). Hence, since G contains 

(D) as we have just seen, it will contain either U or UVU~^ 
in the cases considered. We therefore have the following types: 

(E) Group of order 36</> generated by (C) as given in 113: 

5i = (l, CO, a>2), T: a;i=:;'2, X2=x'3, :r3=x'i, 

and the transformation V of (3). 

(F) Group of order 72<l> generated by Si, T, V and UVIJ-K 

(G) Group of order 216^ generated by Si, T, V and U. 
These groups are all primitive, and they all contain (D) 

as an invariant subgroup. The group (G) is called the Hessian 
group (cf . Jordan, Journal fiir die reine und angewandte Mathe- 
matik, Bd. 84 (1878), p. 209). 

116. On Roots of Unity. A solution of the equation 

^ = 1, 

n being a positive integer, is called a root of unity. A solution 
a 's in particular called a primitive th root of unity, if n is the 
least integer for which a'* = l. In such a case n is called the 
index of the root. 

Theorems. 1. The product or ratio of two roots of unity 
is again a root of unity. 



240 THE LINEAR GROUPS IN THREE VARIABLES (Ch. XII 

2. Any positive or negative rational power of a root of 
unity is again a root of unity. 

3. If n is the index of a root a, and m a positive integer, 
the index of a" is n/d, where d is the highest common factor 
of n and m. 

4. If the index of a root 6 is n = ah, where a and h are two 
integers which are prime to each other, then it is possible to 
find a root of index a, say a, and one of index h, say /3, such that 

As is customary, we write w, w^ for the roots of index 3; 
i, i for the roots of index 4 ; co, co^ for the roots of index 
6, etc. 

5. If a is a primitive wth root, then the n roots of a::" 1 = 
are a, o?, . . . , a"~S a", and we have 

l+a+a2+ . . . +a'-i=0. 

6. Theorem of Kronecker. For the proper handling of a 
certain class of equations we use a very efTective theorem of 
Kronecker.* Instead of making a formal statement of the 
theorem we shall explain its meaning by implication. 

The class of equations referred to are all of the form 
S|lj[ai = 0; ai, . . . , at being roots of unity, and the ques- 
tion involved is this: if these roots are not known originally, 
but their number k is known, what can be inferred about their 
values? The theorem implies that the k roots fall into sets, 
each containing a prime number of roots the sum of which 
equals zero. Moreover, if p be the number of roots in any 
one of the sets, and if a be a root of index p, then the roots of 
the set are e, ea, . . . , ea""^, where e is an unknown root of 
unity. We shall discuss in full the cases ^ = 3, 4, 5. . 

^ = 3: ai4-a2+3 = 0. Here we have a2=aiw, a3=aico^. 

^=4: ai+a2+a3+a4 = 0. We have two sets of two roots 
each, say ai +a2 = 0, as -|-a4 = 0. 

k = 5: ai-|-a2+a3+a4+a6 = 0. There are two possibilities: 
one set only, or two sets containing 3 and 2 roots respectively. 

* Mfimoircs sur les facteurs iiT6ductiblcs de I'expression .t** 1, Journal de 
Alalliemaliqiics pares et appliqu6cs, scr. 1, t. 19 (1854), p. 178. 



117] THE PRIMITIVE SIMPLE GROUPS 241 

If /3 represents a primitive 5th root, and 7, 5 roots of unknown 
indices, the two cases are respectively given by 

7+7/3+7/3^+7/33+7/3^ = 0; 

(7-7) + (5+5w+6co2)=0. 

By means of Kronecker's theorem the following can be 
proved : 

7. If N represents the sum of a finite number of roots of 
unity and k an integer, and if it be known that N'/k is an algebraic 
integer (that is, a solution of an equation xf^-{-aixf^~^-\- . . . 
-\-am = 0, where ai, . . . , a^ are positive or negative integers 
or zero), then N/k equals the sum of a finite number of roots 
of unity. 

More definitely, the roots in N can be arranged into two 
sets such that the sum of those in one set vanishes and those 
in the other set are each repeated k (or a multiple of k) times. 

Primitive Groups whose CoRRESPoisroiNG Collineation 
Groups are Simple,* 117-123 

117. Theorem 13. No prime p>7 can divide the order oj 
a primitive linear group G in three variables. 

Proof. The process consists in showing that, if the order 
g contains a prime factor p>l, then G is not primitive. We 
subdivide this process into four parts as follows: 1 proving the 
existence of an equation F = 0, where F is a certain sum of 
roots of unity; 2 giving a method for transforming such an 
equation into a congruence (mod p); 3 applying this method 
to the equation F = 0; 4 deriving an abelian self-conjugate 
subgroup P of order />*. 

1. The order g being divisible by p, G contains a Sylow 
subgroup of order p^ and therefore a transformation 5 of order 
p. We choose such variables that 5 has the canonical form 

S = {ai, a2, as); ai'=a2'=3*= 1, aia2a3 = l. 

Two cases arise: two of the multipliers are equal, say ai=a2, 

or they are all distinct. They cannot all be equal, since ai'= 1 

* We shall briefly call siu h groui^ primitive simple groups. 



242 THE LINEAR GROUPS IN THREE VARIABLES [Ch. XII 



and ai^= 1 imply ai = 1, whereas S is not the identity. Of the 
two cases we shall treat the latter only: the method would be 
the same in the former case,* and the result as stated in 
Theorem 13 would be the same. 

Selecting now from G any transformation V of order />: 

bi 



V = 



a\ 



Cl 



02 1)2 C2 

as h cs . 



we form the products VS, VS^, VS*". Their characteristics 



( 89) and that of V will be denoted by [VS], 
we have 

[V] =ai +62 +C3, 

[VS] =aiai -\-b2a2 -\-C3a3, 

[VS^]=aiai^+b2a2^-\-C3a3^, 

[75"] =aiai''+&2a2''+C3a3''. 



[V], and 



(4) 



(5) 



= 0. 



We now eliminate ai, &2, cs from these equations, obtaining 
[V] 111 

[VS] ai a2 OL2, 
[F52] ai2 a2^ az^ 
[75"] ai" a2'' aa* 

Expansion and division by (ai a2)(a2 a3)(a3 ai) gives us 

(6) [F5'']+i<:[F]+Z,[F5]+iW[F52] = 0, 

K, L, M being certain polynomials in ai, a2, a3, with the general 
term of the type a.^a'^a^. Since ai, a2, as are powers of a 
primitive />th root of unity a ( 116, 5), the quantities K, . . . 
are certain sums of powers of a. Moreover, the characteristics 
[F] . . . are each the sum of three roots of unity (81, Ex. 7). 
If, therefore, the products in (6) were multiplied out, there would 
result an equation of the kind discussed in 116, 6. The 

* The congruence (10) would here be of the first degree in u- 



117J THE PRIMITIVE SIMPLE GROUPS 243 

various terms could therefore be rearranged in sets as explained 
in that paragraph, which gives us an equation of the form 

(7) ^(l+a+a2+ . . . +p-i)-f.5(i+^_|.^2 , , ^ 4.^-1) 
+C(H-7+72+ . . . +7'-')+ ... =0, 

A, B, C, . . . being certain sums of roots of unity; a, /3, 7, 
. . . primitive roots of the equations x^ = \, a:* = l, c^=\, . . . 
respectively; and p, q, r, . . . different prime numbers. 

The coefficients A, B, C, . . . may be put into certain 
standard forms. Thus, any root cj^l occurring in any of 
these sums will be assumed to be resolved into factors of prime- 
power indices (116, 4): = pet(r . , the root ep being 
of index p*^, e, of index ^, etc. Furthermore, within A any 
root p will be assumed to be either unity or a root whose index 
is divisible by p^. For, if it were of index p, say p=a*, we 
could put 1 in its place, since 

a*(l+a+a2+ . . . +0^-^) =*+*+' +a*+2_^ . . . +a*+'- 

= l+a+a2_|_ _ ^ _|_^-l 

by means of the relation a^ = l. Likewise we assume that 
any root cj within B is either equal to unity or is a root whose 
index is divisible by q^; and so on. 

To illustrate, let i be a root of index 4 and t a root oT index 9 (namely 
'=w), and let p = 2. q=3. Then the standard form for the expression 

(8) ( /- 1)(1-1)+(t^u - w^+i){l+u-\-w') 
would be 

2. We shall now make certain changes in the values of the 
roots in the equation (7). First we put for every root ,, 
cr, . . . whose index 19 divisible by the square of a prime 
other than p^ (as 7^ in the example above), leaving undis- 
turbed the roots whose indices are not divisible by such a 
square, as a, o^, . . . , /3, . . . The quantities A, B, . . . 
are thereby changed into certain sums A', B\ . . . The 
equation (7) is still true, the vanishing sums l+a+o^-\- . . . 
-fo^"*, etc., not having been affected. 



244 THE LINEAR GROUPS IN THREE VARIABLES (Ch. Xll 



Next we put in place of q 2 of the roots /3, /3-, . . . , 
/S*"*, and 1 for the remaining root, thus changing 
5'(l+/3+)32+ . . . +/3-) into J5'(l+0+0+ . . . +(-1)), so 
that this product still remains equal to zero. Similarly, we 
put in place of r2 of the roots 7, t^, . . . , y*, and 1 
for the remaining root, and so on. Proceeding thus, we shall 
ultimately change (7) into an equation of the form 



^"(l+a+a2+ 



+a'-0=0, 



where A" contains roots of the form p only. 

Finally, we put 1 in the place of every root a, oc^, . . . , 
c^~^, as well as every root tp. The left-hand member may then 
no longer vanish, but will in any event become a multiple of p. 

The final value of the expression (8) would be (w + l)(l+l) = 2 or 0, 
according as is replaced by or L 

Notation 1. Any expression N which is a sum of roots of 
unity, changed in the manner described above, shall be denoted 
by N',. 

3. We shall now study the effect of these changes upon 
the left-hand member of (6). Each of the characteristics 
[VS], . . . , [VS'^], being the sum of three (unknown) roots 
of unity, will finally become one of the seven numbers 0, 1, 
2, 3, whereas [V], being the sum of three roots of index 
1 or /> (cf. 1), will become 3. The left-hand member of (6) 
will thus take the form 



(9) 



[VS'^yp-\-3K'p+L'AVSY+M'AVS^Y 



and this number is a multiple of p (by 2). 

The values K'p, L'p, M'p may be obtained by treating them 
as indeterminates 0/0. Thus, 



K = 



1 



(ai a2)(a2 a3)(a3 ofi) 



ai 2 as 
ai^ ai^ a-^ 
ai** 02** a.s** 






We find 

i^'p=-KM-l)(M-2), r, = M(M-2), 3/',= -lp(M-l), 



117] THE PRIMITIVE SIMPLE GROUPS 245 

and if we substitute in (9) and multiply by /> 1 we obtain 
the congruence: 

(10) [VS''Yp=sn^-\-tn-\-v (mod/), 

s, t, V being certain integers, the same for all values of ft. 

We finally substitute in succession n = 0, 1, 2, . . . , p \ 
in the right-hand member of (10). The remainders (mod p) 
should all lie between 3 and +3 inclusive, the interval of the 
values of [F5'']'p. Now, each of these seven rem.ainders can 
correspond to at most two different values of n less than p, if s 
and / are not both =0 (mod p), by the theory of such congru- 
ences. Hence, there will correspond to the seven remainders at 
most 14 different values of n, so that p is not greater than 14 
unless s=t = 0. Tr>'ing p = 13 and P = ll, choosing for s, t, v 
the different possible sets of numbers </> (the problem can be 
simplified by special devices)* we find that in no case can the 
remainders all be contained in the set 0, 1, 2, 3, unless 
5 = /^0. Choosing therefore this alternative we get, if P>7, 

[VS'^Yp^v (modp). . 
In particular, 

[VS]'p^vMVYp = Z (modp), 

from which it follows that [F5]'p = 3. Again, from this equation 
we deduce that the roots of [VS] are of index 1 or p. For, 
if the index of one of these roots were divisible by the square 
of a prime, or by a prime different from p, then the changes 
indicated in 2 could be made at the outset in such a way that 
or 1 would take the place of this root. But then [F5]'p. 
would be one of the numbers 0, 1, 2, 3. 

4. Accordingly, the product VS of any two transforma- 
tions both of order ^ is a transformation of order ^ or 1. The 
totality of such transformations in G, together with E, will 

* Since an-\-b runs through the p values 0, 1, 2, ... , ^1 (mod p) when 
fi does, we may substitute this expression for n in the right-hand member of (10) 
and select constants a,b so that this member takes a simpler form. For instance, 
if P=ll, the right-hand member of (10) may be reduce<l by this substitution 
to one of the forms M*+c; m; or c; according as s^O; ssO, tj^O; smImO 
(mod p). When p= 13 we get the forms m*+c; 2mHc; m; c. 



246 THE LINEAR GROUPS IN THREE VARIABLES [Ch. XII 

therefore form a group P. The order of this group must be 
a power of />, since it contains no transformations whose order 
differs from /> or 1. Moreover, P is invariant in G, since an 
operator of order p is transformed into one of order p. Hence, G 
has an invariant subgroup P of order p*. But this subgroup 
is abelian ( 108, Corollary) and therefore G is intransitive or 
imprimitive ( 108, Lemma). 

Notation 2. A quantity iV, which is the sum of a certain 
number of roots of unity, in which every root e,, is replaced by 
1, but in which none of the other changes indicated in 2 are 
carried out, will be denoted by Nj,. If iV = 0, then Np = 
(mod p). 

118. Theorem 14. If a group G contains a transformation 
S of order p^<t), P being a prime >2, then there is an invariant 
subgroup Ep in G (not excluding the possibility G=Hp) which 
contains S^. Any transformation in H,, say T, has the property 
expressed by the following congruence: 

(11) [V]MVT], (modp), 

V being any transformation of G. 

In the case p = 2 the group G contains an invariant subgroup 
Hp if the order of S is p^, and S^* will belong to Hp,- also if 
S = { l,i,i),in which case S^ will belong to Hp. 

The proof follows the plan of that of the previous theorem. 
If p> 2, we write S in canonical form, and construct the products 
VS, VS^, VR, where R denotes 5*. Assuming that the three 
multipliers of 5 are all distinct, we obtain an equation cor- 
responding to (6) in 1: 

[VR]-\-K[V]-\-L[VS]-\-M[VS^]==0. 

However, the changes indicated in 2 are not carried out except 
that 1 is put for every root e, whose index is a power of p (cf. 
Notation 2 above). The coefficients L, M become multiples 
of p by this change, and we find K= \ (mod p). Hence 
finally, 

[VR]p-[V]p^O (modp). 

Now consider all the conjugates Ri, . . . , 7?^ to i? within G. 
They generate an invariant subgroup H, ( 14, Ex. 6), and 



1191 THE PRIMITIVE SIMPLE GROUPS 247 

they all have the property of T as expressed by (11), since they 
fulfil the conditions of the theorem. Moreover, any trans- 
formation T in //j, satisfies the congruence (11), since such a 
transformation can be written as a product of powers of 
Rif . . . y Rh. For instance, let T = RiR2, and we have 

[VRi]MV]v, [(VRi)R2]MiVRi)]p {modp). 
Hence, 

[VTl = [VR^R2]MVRl]MVV 

119. The Invariant Group Up. The order of this group is a 
power of p. For, if its order contained a prime factor q, 
qj^p, there would be a transformation of order q in Hj,, say 
T. Then, if E represents the identical transformation, we have 
by (11), 

[Vl = [EV]ME]v=^ (modp). 

Hence, the multipliers of T being a, 0, y, we have 

a^+/3^+y=3 (mod p), 
and therefore 

(12) 2"V+^'+V)^32)--^. 

Unless T is a similarity-transformation (which we may assume 
it is not), the roots a, /3, y are not all equal. Hence, the left- 
hand sum is 2q or q according as a is or is not equal to one of the 
roots /3, y. The right-hand sum is, however, Sq or zero, accord- 
ing as a is or is not unity. It follows that the congruence is 
impossible except when p = 2, and a is one of the roots /3, y, or 
unity. Substituting ^~^ and y~' for a"-' in (12) we get similar 
results. Collecting these, we finally discover that in no case 
can q^^p. 

The order of Hp is accordingly a power of p, and the group 
is monomial (Theorem 10). The possibility G = Hp is accord- 
ingly untenable if G is primitive. 

Corollary. No primitive simple group can contain a Irans- 
formation of order p^<t> if p>2\ or p^ if p = 2. 



248 TIIK I IMAR GROUPS IN THREE VARIABLES [Ch. XII 

120. Theorem 15. No primitive simple group can contain 
a transformation S of prime order p, p>3, which has at most 
two distinct multipliers. 

Let S = {a\, ai, a2) * and assume first that /> = 7. This 
transformation leaves invariant a point (a:i=X2 = 0) and every 
straight line through it. This will also be the case with any 
other transformation S' conjugate to S (Theorem 3). There- 
fore, the line joining the two invariant points is invariant 
for both S and S'. If now the variables be changed so that 
the common invariant line is yi = 0, the group generated by 5 
and 5' will be reducible and therefore intransitive, breaking 
up into a group in one variable (3'i) and one in two variables 
(>'2, >'3). But, there being no primitive or imprimitive finite 
linear groups in two variables generated by two transforma- 
tions of order p = l (cf. Chapter X), it follows that S and S' 
are commutative. 

Accordingly, all the conjugates to S are mutually com- 
mutative and generate an abelian group, which must be invari- 
ant in G (14, Ex. 6), and the latter cannot be primitive 
(108, Lemma). 

Next, let /> = 5. If 5 and 5' are not commutative, they gen- 
erate the icosahedral group (E), 103, in the variables y2, ya. 
This contains a transformation of order 3 whose multipliers 
are w, (j?, and a similarity-transformation whose multipliers 
are 1, 1 (cf. 97). The product of these two transforma- 
tions, as a transformation in the variables yi, y2, ya, can be 
written in the canonical form T = {\, w, oP). But such a 
transformation is excluded by the next theorem [put Si = 
r2 = (l, 0)2, co) and52 = r3 = (l, -1, -1)]. 

121. Theorem 16. No primitive simple group can contain 
a transformation S of order pq, where p and q are different prime 
numbers, and Si = S^ has three distinct multipliers, while 52 = 5" 
has at least two. 

Let 5 be written in canonical form and assume 5i = (ai, a2, 
as). We then construct the transformations V, VS2, VSi, 
VS\^ and proceed as in 117, 1, obtaining an equation corre- 

* Such a transformation is called a homology. 



122] THE rUIMITIVE SIMPLE GROUPS 249 

spending to (5). After putting unity for every root of index 
/>*, the equation becomes the congruence: 

\[VS2]p-[V]j>\{oci-a2){a2-a3){a,i-ai)=0 (mod p), 

which can be changed into the following: 

{[vs2l-[vuf^o 

after multiplying by a suitable factor, since 

(l-6)(l-62) . . . (l-e'-')=lim.^^=9 

z-l Xl 

when is a primitive qth root of unity. 
Hence finally, 

[VS2]MVl (modp), 

and the argument of 118 is now valid. 

Corollary. No primitive simple group can contain a 
transformation of order 35, 150, or 21 <^. (If Si, representing 
respectively 5^, S^*, or 5^*, has not three distinct multipliers, 
Theorem 15 applies.) 

122. The Sylow - Subgroups. Consider now a primitive 
group G of order g<t). A possible subgroup of order 5^ or 7^ 
would be abelian ( 108, Cor.). By trial we find readily that 
no such group can be constructed without violating Theorem 
15 or the Corollary to Theorem 14. Again, if g is divisible by 
35, we have a transformation of this order ( 135, Cor. 3). 
But this is impossible in a primitive simple group ( 121, Cor.), 

A subgroup of order 3*<^ is monomial, and contains an abelian 
subgroup of order 3*~*0 at least. Assume ^ = 3; we then have 
an abelian subgroup P' of order S^<f>. When we construct 
such a group, avoiding the invariant subgroup Hj, resulting 
from Theorem 14, we discover that it must contain a trans- 
formation of type T = (i, , (oi^), where ^ = w. Then if g is 
divisible by 5 or 7 at the same time, we would have a trans- 
formation of order 15</) or 21<^ ( 135, Cor. 3), violating the 
Corollary, 121. In any event, we can have no abelian sub- 
group of order 3*"^ if *>3 (cf. E.x. 4, 109). 

Finally, a subgroup of order 2* is monomial and contains 
an abelian subgroup of order 2*"* at least. By trial we find 
thatiSj-U2(cf. Ex.3, 109). 



250 THE LINEAR GROUPS IN THREE VARIABLES (Ch. XII 

Collecting our results, we find that g is a factor of one of the 

numbers 

23-33, 23-32.5, 23-32.7, 

and the question arises what simple groups can have such 
orders. 

Now, all simple groups whose orders do not exceed the largest 
of these numbers have been listed. There are four possi- 
bilities: 

g = 60, 360, 168, 504. 

123. The Three Types of Primitive Simple Groups. The 
first two of the numbers just given are the orders of the alter- 
nating groups on 5 and 6 letters, and the last two the orders 
of certain transitive substitution groups on 7 and 8 letters 
(20). The last case may be excluded from consideration 
for the reason that we should here have a Sylow subgroup of 
order 8 which is abelian and generated by three operators 
each of order 2 (20, Ex. 3). But no such group can be con- 
structed in three variables, as may be found by trial. 

In the other c^ses the corresponding types may be con- 
structed after a set of generators and their generational rela- 
tions are given (cf. 103, and Ex. 2, ibid.). The variables are 
selected so that the generators appear in as simple a form as 
possible; for instance, some one of them is always written in 
canonical form at the outset. We shall omit the details 
here; it would afiford excellent practice for the student to carry 
out this work. 

(H) Group of order 60 generated by 1, 2, 3, satisfying 
the relations:* 

i3=22=32 = l, (i2)3 = (2^3)3 = 1, (i3)2 = l; 

namely: 

1: X\=X2, X2=X3, ^3=i; 

2 = (1, -1, -1); 

3: Xi=^(-x\-\-H2x'2-\-Hlx'3), X2 = Ut^2x\-\-Hix'2-x'3), 
X3=^inix'i -x'2-{-fJi2x'3) ; 

* E. H. Moore, Proceedings of the London Mathematical Society, vol. 28, p. 357. 



124] INVARIANT PRIMITIVE SUBGROUPS 251 

where 

(I) Group of order 360<^, generated by i, 2, 3 of (H) and 
4, where 

42 = 1, {ExE^Y = {E2E,Y = {E'iE^f = 1 ; 

4: :ri = ic'i, X2=0)X'3, X3= a)V2. 

(J) Group of order 168 generated by 5, T, R with the 
relations (cf. 20, Ex. 2) 

57 = ^3 = 2^2^1, T-^ST=S\ R-^TR = 'n, {RSy = l; 
namely: 

S = (^, /32, /34); 

T: Xx=x''2., X2=x'z, X2,=X'\\ 

R: xi=h(ax\-\-bx'2-\-cx'3), X2 = h{bx\-{-cx'2-^(ix'3)) 
Xz = h {ex' i+ax'2+hx'z); 
where 

1 



/r=-^(^+^2_^^_^6_^5_^3) = 



V37- 



124. Primitive Groups Having Invariant Primitive Subgroups. 
We have already determined the primitive groups containing 
an invariant subgroup of order 2^ or 3^ (114-115). The 
possibility of a subgroup Hj, shall therefore be excluded from 
consideration (cf. 119). 

Let us assume that the group (H) is contained invariantly 
in a larger group (H') of order GO/f</). In (H) we have 6 sub- 
groups of order 5, which must be permuted among themselves 
by (H'). One of them must therefore be invariant in a sub- 
group of (H') of order 6O/?0/6 = 10//(^. But no such group can 
be constructed if h> 1 without introducing the group lip. 

In this manner the cases (I) and (J) may be disposed of. 
There are left the cases (E), (F), (G). Now, these groups 
all permute among themselves a single set of four triangles. 
This set is therefore also permuted among themselves by a 
larger group including either (E), (F) or (G) invariantly. But 
the groups having this property were found in 115 to be just 
these three. 



252 THE LINEAR GROUPS IN THREE VARIABLES [Ch. XII 



EXERCISES 

1. Determine the linear groups in two variables by the method of 
this chapter. (To determine the primitive simple groups, show first that 
the order of such a group is a factor of 60<^). 

2. Prove by the method of 120 that a primitive group in four vari- 
ables cannot contain a transformation whose order is greater than 5, 
if its characteristic equation has only two distinct roots. 

3. Why is it necessary to add the statement " while 52=5* has at 
least two " (distinct multipliers) at the end of Theorem 16? 

4. Obtain the group (H) by the method of Ex. 2, 103, in the following 
form: 

S={abcdc) = (l,e\ c); 

U=(ad){bc): a;i= 3c'i, a!;2= x's, X3=x'i', 

T=(ab)icd): Xi=-^ix\+x'2+x'z), 
V5 

2 = -7^(2:>;'i +sx'2 +tx'z) , Xz = p(2x\ -fte'j +sx',) ; 
V5 V 5 

where 

e5=l, S=e^+e\ t = +e*, Vl = t-S. 

Show also that (E) as given in Ex. 2, 103, transforms the variables 
yo= XiXi, yi = X2'^, T2= Xi^ into linear functions of themselves and hence 
appears as a linear group in three variables, which is precisely the group 
(H) as just exhibited if wc write yo, yi, yz for Xi, x^, Xz respectively in (H). 

5. Obtain the group (I) by adding a transformation W={ad)(ef) 
to the list S, U, T of (H), Ex. 4, and show that 

W: Xi = ^(x\+\ix'2+\ix',), X2=-j:,(2\2x'i-\-sx\-\-txf,), 
V5 v5 

Xz=--.{2X,x'i+lx'2+sx't); 
v5 



where 



Xi = i(-lV"l45), X2 = i(-lTV^^). 



Tn this and the previous example we need not verify that the transforma- 
tions obtained actually generate the respective groups as soon as they 
have been fully determined from a number of arbitrarily chosen relations, 
since just one type of each group (H) and (I) has been shown to exist by 
the generational relations given in 123. (In determining W, account 
has to be taken of the fact that (I) contains similarity-transformations; 
cf. the determination of (E), 103.) 



125 INVARIANTS OF GROUPS IN THREE VARIABLES 253 

126. Invariants of the Linear Groups in Three Variables. 

The theory of these invariants is by no means so simple as the 
theory of the invariants for the groups in two variables, and 
it is beyond the scope of this book to enter into a general dis- 
cussion, except as to indicate the principal results. 

In all cases we have a set of fundamental invariants such 
that all others are rational integral functions of these.* We 
shall here give such sets for the groups (G), (H), (I) and (J). 
That they are invariants may be verified directly by the student. 

(G) If we introduce the abbreviations 

XiX2X3 = <f>, Xi^-\-X2^-\-X3^ = rp, Xi^X2^ -\-X2^X's^-\-Xs^Xi^ = X, 

the fundamental set consists of the functions f 

C6 = ^-12x; 

Cq = {xi^ X2^) (x2^ - X3^) {xs^ Xi^) ; 
Ci2 = ^(^ +21603). 

/)l2 = ^(2703-^3). 

which satisfy the relation 

{432Cg^-Ce^+SCeCi2y = 4(172SDi2^-\-Ci2^). 
(H) Here we have four fundamental forms. One of them is 

A =Xi^+X2X3, 

if we take the group as given in 124, Ex. 4. 

The other three can be obtained in a manner similar to that 
which we employed in the case of the group (E), 105. We 
select a linear invariant /i for each of a set of subgroups of 
(H) of orders 10, 6, 4, respectively, and form the products of 
the 6, 10, 15 different linear expressions into which /i is trans- 
formed by (H). Selecting, for instance, the group generated 
by S and U, Ex. 4, 124, for the subgroup of order 10, we get 
fi=xi and a corresponding invariant 

Bl=Xl{Xl-\-X2+X3){Xi+i'^X2-\-(X3){xi-\-X2+f*X3) 

' {xi -\-^X2+e^X3) {xi -^e^X2 -h^Xs) 
= Xi (0:2* +a;3* +5^:1X2^X3^ 5:ri%2:JC3 +iCi*) 

* Cf. Dickson, Algebraic Invariants (Wiley & Sons), New York, 1914, p. 70 ff. 
t Maschke, Mathemalische Annalen, Bd. 33 (1889), pp. 32.'>-.326. 



254 THE LINEAR GROUPS IN THREE VARIABLES ICh. XII 



These invariants must evidently be transformed into the 
functions/, E^ T of the group (E), 105, by the substitution 
of X2X\, X2^j Xi^ for xi, X2, xz respectively (cf. Ex. 4, 124). 
The relation among the invariants A, B\, . . . corresponding 
to that connecting /, H, T is of degree 30 in the variables Xi, 
X2, X3, and its determination may be left as an exercise for the 
student.* 

(I) The fundamental invariants are four in number, f 
One is the function 

F=A^-\-\Bu 

where (taking for (I) the group given in Ex. 5, 124) ^4 and Bi 
are the functions listed above under (H), and 



X = 



9dz3V-15 
20 



The three other invariants are obtained by evaluating the 
Hessian of F: 





Fn 


F\2 


Fl3 




x= 


F21 


F22 


F22. 


f 




F31 


F32 


F33 




then the bordered Hessian : 










Fn 


F12 


Fi3 


Xi 


F = 


F21 
Fai 


F22 

Fs2 


F23 
F33 


X2 
X3 




X, 


X2 


X3 






and finally the Jacobian of F, X, and Y. 

(J) As in the previous case, the fundamental invariants 
are four in number; J a function 

J=Xt?X3 -\-X2^xi +:*:3^iC2 ; 

the Hessian of / (say X) ; the Hessian of / bordered by the 
first derivatives of X (as Y above); finally, the Jacobian of 
the three functions already determined. 

* Cf. Klein, Vorlesungen Uher das Ikosaeder, Leipzig, 1884, p. 219. 
t Wiman, M alhemalische Annalen, Bd. 47 (1896), pp. 548-556. 
i Klein, M alhemalische Annalen, Bd. 14 (1879), pp. 446-448. 



126) ORDER OF A PRIMITIVE GROUP 255 

EXERCISES 

1. Construct a fundamental set of invariants for the groups (E) and 

(F). 

2. The invariant Dm under (G) is the product of the four functions 
U, k, ti. It, 113. Are there other relations among the invariants C, 
Ca, Ci2, Di2 and the functions /i, h, tz, i*? 

126. Order of a Primitive Group in u Variables. In con- 
clusion we shall state some general results bearing on the order 
of a primitive group G in w variables. 

The Theorem 13 admits of generalization to n variables as 
follows. The number of the equations (4) would now be + l, 
the left-hand members being [V], . . . , [75""^] and [75"]. 
EHminating ai, 62, . . . , we obtain the corresponding equa- 
tion (5), from which a congruence similar to (10) is derived: 

[r5'']'p=5M''-'+/M'*"^+'t'M''"'+ . . . (mod p). 

In this we substitute in succession )u = 0, 1, . . . , p 1. The 
left-hand members are integers having the 2w-|-l values rang- 
ing from n to +w inclusive, and for each such value we can 
have at most 1 values of n by the theory of congruences, 
unless the right-hand member is merely a constant. Hence, 
excluding this possibility, there are at most ( l)(2w-f-l) 
values of n corresponding to the values assumed by the right- 
hand members; accordingly, if p> {n l){2n-\-i), the right- 
hand member is a constant merely, or 

[VS'^YMVYp^S (modp). 

From this it follows as in 117, 4, that the group has an invari- 
ant abelian subgroup of order />* and cannot be primitive. 
Hence, tite order of a primitive linear group in n variables can- 
not be divisible by a prime greater than (w 1)(2-|-1). 

The Theorem 14 may similarly be generalized to read: 
if a group G contains a transformation S of order ^4>^pn(t>, 
then there is an invariant subgroup lip in G whose transformations 
have the property 

[V]MVrl (modp), 

where V and T represent any transformations of G and H, respect- 
ively. 



)r.) and its order is ^5"^ 
emaining factor of g is the 
50 ^5"-^ It follows that 
variables is 



oved the classical theorem that 
ablcs is of the form Xf, where / 
ibgroup, and where X is inferior 
upon n (Journal fiir die reine 
Definite limits to X have been 
laracteristics belong to a given 
oniglich Preussischen Akademie 
Jieberbach and Frobenius {Sitz- 
fT); and by the author {Trans- 
ety, vol. 5 (1904), pp. 320-321 
5 for imprimitive groups). The 



CHAl 

THE THEORY OF G 

127. Introduction and Del 
by Frobenius and is largely 
Burnside | simplified the th( 
tions. We shall devote thii 
main points of the theory 



258 GROUP CHARACTERISTICS ICh. XIII 

A function /(xi, . . . , x) which is transformed into a con- 
stant multiple of itself by every transformation of G is called 
an invariafit of G. It is an absolute invariant if the constant 
multiplier is unity for every transformation of G; otherwise 
it is a relative invariant. 

A series of invariants /i, - - - , ft are said to be independent 
of each other if the variables xi, . . . , Xn cannot be eliminated 
from the equations 

fi=ai, . . . , /t = a*, 

where ai, . . . , ai are arbitrary constants. They are said to 
be linearly independent if no identity exists of the form 

61/1+ . . . +Vt=o, 

where bi, . . . , bt are constants, not all zero. 

We shall as hitherto denote by (f)S the result of operating 
upon a function / by a transformation S. 

128. Theorem 17. The number of linearly independent 
absolute invariants of G of the first degree in xi, . . . , Xnis 

it- 

t-i 

Proof. 1. If Sxt5^0, then G will have an absolute invariant 
of the first degree. For, let^'i, . . . , yn be arbitrary constants, 
and let/=yia;i+ . . . +ynXn. Then the function 

(/)5i+ . . . -\-{f)S,^F 

will be an absolute invariant of G provided that it does not 
vanish identically. This is seen as follows. We have 

(F)5, = (/)5i5,+ . . . +(/)5^r = (/)5i+ . . . +(f)S,=F. 

Thus, F is an absolute invariant unless it vanishes. If 
it does, then 2xi = 0. For, the sum of the coefficients of yixi, 
y2X2, ' , ynXn in {f)St IS readily found to be xt- Hence, 
at least one of these terms will be present inF if 2x5^0; say 
yixi. If therefore we put yi = l, ^2= . . =yn=0, the func- 
tion F will contain a term involving xi and hence it does not 
vanish. 



128] 



NUMBER OF INVARIANTS 



259 



2. Let us now suppose that G has just k linearly inde- 
pendent absolute invariants of the first degree. Wc may 
assume that the n variables were chosen originally such that 
xi, X2, . . . , Xk are the invariants in question. Then G 
is intransitive, breaking up into ^ + 1 sets of intransitivity, 
containing respectively 1, 1, . . . , 1; n k, variables: Xi, 
X2, , Xk-, (Xk+i, . . . , Xn), by Theorem 6. 

Thus, if M = 2, ^=1, and if Xi is the absolute invariant, the matrix of 
any transformation of G will be of the form 

1 o" 

c d 

The reducible group may now be transformed into an intransitive group of 
the following special form 

'l 
d 

The characteristic xiSt) is accordingly equal to k-\-x(S't)y 
where S't is the transformation corresponding to St in that com- 
ponent of G which involves the set (a:t+i, . . . , Xn). Now, 
Ttx{S\) =0, or we would have a new invariant by 1. Hence, 

which completes the proof. 

Corollary 1. The number of linearly independent abso- 
lute invariants of degree m in xi, . . . , Xn is 



jX^(sn, 



where x(Si^^^) represents the sum of the homogeneous products 
of degree m in the multipliers of St, namely 

ar+a2'"+ . . . +ai'""^a2+ . -\-ar-^a2a3-\-. . . 
Proof. For brevity we take n = m = 2. When the variables 
Xi, X2 are subjected to a linear transformation 

b] 
S = 



260 GROUP CHARACTERISTICS [Ch. XIII 

the products of x\, x-^ of the second degree, namely Xi^, xiX2< 
X2^, are correspondingly subjected to a linear transformation 



5< = 



a2 2ab b'' 
ac ad-\-hc bd 
c2 2cd d^ 



and to a group G of transformations S, . . . will correspond 
an isomorphic group G^^ of transformations S^^\ .... If 5 
is written in canonical form {a, /3), so is S^^\ namely (o^, ajS, /S^). 
Hence xC-S^^^) =^+a,5+/32. 

We now apply Theorem 17 to the group G^^^ and obtain 
the Corollary 1 for the case n = m = 2. 

Corollary 2. In the case of a transitive group, 11lZlxt = ^^- 
For, in this case there are no invariants of the first degree. 

By an elaboration of this principle we obtain the important result 
that if *< represents any given integral symmetric function with integral 
coefficients of the multipliers of St, then 2j<Ii*=/g, where / is a positive 
or negative integer or zero.* 

EXERCISES 

1. Write a given substitution group as a linear group so that the 
letters of the former are the variables of the latter, and determine the 
characteristics. 

2. Prove that the average number of letters which remain unchanged 
by a substitution of a transitive substitution group G is equal to unity. 
(Prove first that G contains a single absolute invariant of the first degree.) 

129. Lemma. Let there be given a function 

/=ZiFi+ . . . +XtYt, 

where Xi, . . . , Xt are linear functions oi xi, . . . , Xn, 
the variables of a group G', and Fi, . , . ,Yt linear functions of 
yi, . . . , ym, the variables of a group G" simply isomorphic with 
G'. We may assume that Xi, . . . , Xt (as well as Fi , . . . , Ft) 
are linearly independent of each other; if it were possible to 

* See Transactions of the American Mathematical Society, vol. 5 (1904), p. 
.464ff. 



130] 



TRANSITIVE GROUPS 



261 



write, say, Xk = aiXi-h . . . +at-iA'*_i, the function/ could 
be expressed in fewer than k terms: 

A'i(Fi+aiFt)+X2(F2+a2Ft)+ . . . + Xt-iiYt-i+at-iYtY 

= XiF'i+X2F'2+ . . . +Xt_,F'*-i. 

Now if f be unaltered when operated upon simultaiuously 
by corresponding transformations of G' and G" (in otlier words, 
if f is a bilinear invariant), and if k<n, G' is intransitive. 

Proof. For the sake of simplicity take ^ = 3. The variables 
of G' and G" may be so chosen tlndii f =x\yi-\-X2y2-\-xzy:i. 

A transformation of G' will change x\, X2, x^ into three 
linearly independent functions of a;i, . . . , Xn, and the corre- 
sponding transformations of G" will change yi, y^, >'3 into 
three hnearly independent functions of yi, , . . , y^. Let 
the resulting expression be /' = X'lF'i + X'2F'2 -f X'sF's, 
and we should have /=/' But this implies that X'\, X'2, 
X'3 are linear functions of xi, X2, X3. Hence, if w>3, G' is 
reducible and accordingly intransitive. 

130. Theorem 18. Representing by x the conjugate-imag- 
inary of Xt we have, for a transitive group G, 



V 



UXt = g. 



Proof. Let G be th_e conjugate-imaginary group of G 
(92). Then, if 5 and S are corresponding transformations 
of G and G: 

6 ... 1 (ab.. 



5 = 



d . 



S = 



the n^ products xiX\, xiXj," . . ^ , a:^x are subjected to a corre- 
sponding linear transformation S' belonging to a group K 
isomorphic with G and G: _ 

' aa ab . . 



S' = 



ac ad . . 



262 GROUP CHARACTERISTICS [Ch. XIII 

The characteristic of S' is the product of the characteristics 
of 5 and S: x{S') = x{S)x{S) (this is seen readily when S (and 
therefore 5 and S') is written in canonical form). Hence, by 
Theorem 17, the number of linearly independent absolute 
invariants of the first degree in the variables of K is Tixixt/g. 
Any such invariant can be thrown into the form 

f=XlXl+X2X2+ . . . +X^, 

where Xi, . . . , X are linear functions of o;;!, . . . , Xn. 

We know one such invariant already, namely the Hermitian 
invariant ( 92), and we may assume the variables originally 
so chosen in G and G that this invariant is 

I=XiXl-\-X2X2-\- . . . +0C^. 

Then, if X be any constant, the expression 

f-h}J=(Xl + 'KXi)Xi-\-{X2-\-\X2)x2+ . . . +{Xn-^'^n)^ 

is also an invariant. 

Now, the constant X may always be determined such that 
Xi+Xrri, X2+Xa;2, . , X+X:r are not linearly independent. 
Therefore either G is intransitive by the lemma above, or 
/+X/ vanishes identically. Hence, since the first alternative 
violates the assumption of the theorem, any invariant / of 
K is merely a constant multiple of / (viz., /= X/); in other 
words, the number 2 x<Xt/g of linearly independent invariants 
/ is unity. The theorem follows. 

EXERCISE 

Prove that if G is intransitive, ^xtxt=lg, where lis a. positive integer 
greater than 1. 

131. Equivalence. Two simply isomorphic groups are 
equivalent if a suitable change of variables in one will make 
the matrices of their corresponding transformations identical. 
If no such choice of variables is possible, the groups are non- 
equivalent. 

For example, the groups generated by the transformations 

r 



5= 



1 



T=il, -1) 



131] EQUIVALENCE 263 

are equivalent. If we put yi = Xi+xi, 3'j = xi jcj in S, then this transfor- 
mation takes the form (1, 1) in the variables yi, y,. 

Theorem 19. Let G' = {S'uS'2, . . . , 5',) andG" = (S"i, 
S"2, , S"g) be simply isomorphic linear groups in n and m 
variables respectively, and lei G' be transitive. Then if S"t repre- 
sents the conjugate-imaginary of the transformation S"t, 



X 

t-1 



x{s\)x{s";)=kg, 



where k = or a positive integer. If k = l, G" is equivalent to G'; 
if k>l, G" is intransitive, and in this case k of its sets of intransi- 
tivity are transformed according to k groups each equivalent to G'. 

Proof. Let x\, . . . , Xn be the variables of G' and y\, 
. . . , 3'fl, those of G". We construct the group K in the nm 
variables Xiyi, . . . , xym, where yi, - - - , ym are the con- 
jugate-imaginaries of the variables of C. Applying Theorem 
17 we find k linearly independent absolute invariants of K 
all of the form anxiyi-j- . . . -{-anmXnym- By a suitable 
change of variables in G" we now cause one of these invariants 
to become xiyi-\-X2y2-\- . . -\-Xnyn, and comparing this with 
the Hermitian invariant xi:ri+ . . . -i-XnXn of G' we may 
readily prove that G" transforms the variables yi, - . , yn 
among themselves according to a group equivalent to G\ Hence, 
if ^ = 1 and m = n, G' and G" are equivalent; if ^^ 1 and m>n, 
G" is reducible and therefore intransitive. 

Conversely, if G" is known to break up into sets of intrans- 
itivity, k of which are transformed according to groups which 
are equivalent to G', then the conjugate-imaginaries of the 
variables yi, . . . , y of any one of these sets will combine 
with xi, . . . , Xn to form one invariant Xiyi-\- . . . + Xiiy^ 
for K, making k such invariants in all. 

Corollary 1. If G' and G" are equivalent, their correspond- 
ing characteristics are equal, and 

x(5'i)-x(5^)+ . . . +x(5'a)-x(5^.)=g, 
if they are non-equivalent and are both transitive, this sum vanishes. 



264 GROUP CHARACTERISTICS (Ch. Mil 

Corollary 2. Let G be a transitive linear group of order g 
in n variables, and let 17 be a regular substitution group ( 27) 
on g letters simply-isomor phic with G. Then if II be looked 
upon as a linear group in g variables, it is intransitive, breaking 
up into a number of component groups among which are found 
just n which are equivalent to G. 

Proof. When a substitution Tt of H other than the identity 
is written in matrix forrti as a linear transformation, every 
element in the principal diagonal is zero, since otherwise the 
corresponding letter would be replaced by itself in Tt. Accord-, 
ingly, x(3^t)=0 unless Tt is the identity; if Ty is the identity: 
(1, 1, . . . , 1), then x{Ti)=g. The transformations of G 
being correspondingly St and 5i = (l, 1, . . . , 1), we have 
therefore 



(-1 



x{St)-x{Tt)=ng. 



The corollary now follows by applying Theorem 19. 

132. Remark. The propositions of 131 become wider in 
scope by an obvious extension of the concept " group." A 
group G' of order g' to which another group G of order g = hg' 
is multiply isomorphic may be exhibited in such a way as if 
it were a group simply isomorphic with G, namely by repeating 
each of its transformations // times. For instance, the sub- 
stitution group of order 6: 1, {ab), (ac), (be), (abc), (acb) is 
multiply isomorphic with two of its subgroups: 1; and 1, (ab). 
With the concept of " group " extended as indicated above, 
we may exhibit the three groups as simply isomorphic in the 
following manner: 

(ac), (be), (abc), (acb); 

1, 1, 1, 1; 

(ab), (ab), 1, 1. 

EXERCISES 

1. Prove that if the regular substitution group H is broken up into 
its ultimate sets of intransitivity with their corresponding componeni 
groups, and ii H he multiply isomorphic with a transitive linear group 6 



1, 


(ab), 


1, 


1, 


1, 


(ab), 



133] 



SUM OF MATRICES 



265 



in n variables, then n of the comiwnent groujjs into which // breaks up are 
equivalent to G. 

2. The group // of order g is always multiply isomorphic with the group 
consisting of the identity alone, which is a transitive linear group in one 
\ ariable: x=x'. Hence, one of the sets of intransitivity of H will contain 
one variable, and the component group will consist of the identity repeated 
g times. Prove this in another way by showing that // possesses a single 
absolute invariant of the first degree. 

Prove also that if an additional set of intransitivity of H contains one 
variable, then H possesses a relative invariant ( 127) of the first degree. 
In such a case H is not a simple group; all those of its transformations 
for which this invariant is absolute form a self-conjugate subgroup. 

3. Among all the component transitive linear groups into which H 
breaks up, let there be k which are non-equivalent: Gi, Gi, . . . , Gi in 
i, 2, . . . , Hjt variables respectively. Prove that g=i^+n2''+ . . . 

133. The Sum of Matrices. The sum of a series of square 
matrices of the same order is the matrix whose elements are the 
algebraic sums of the corresponding elements of the given 
matrices. Thus, 



'ai 


hi 


+ 


'^2 


h2 


^ 


Cl 


di. 




C2 


d2. 





Ci-\-C2 di-\-d2 

li Si, S2, . . are linear transformations in the same vari- 
ables, we shall write ^i +52+ ... to denote the matrix which 
is the sum of the matrices of 6*1, 52, . . . 

Multiplication of matrices is carried out according to the 
rule given in 78, irrespective of whether the matrices repre- 
sent hnear transformations or not. 

We find 

Si +52 =52 +5*1, 

nSi+So-h . . . ) = r5i+r52+ . . . , 

r-K5i+52+ . . . )T = T-'SiT-{-T-^S2T+ . . ., 

(5i+52+ . . . )(ri+r2+ . . . ) = 

5iri+5ir2+52ri+52r2+ . . . . 

If M represents a matrix, and c a constant or a variable, 
the s>Tnbol cAf shall represent the matrix obtained by multi- 
plying every element of M by c. 



266 GROUP CHARACTERISTICS (Ch. XIH 

134. Lemma 1. If Si, . . . , Sm are the different trans- 
formations of a conjugate set of a transitive linear group G in n 
variables, then the matrix 

M=Si+ ... +5. 

is commutative with every transformation of G and has the form of 
a similarity-transformation {a, a, . . . ,a), where a = mx{S\)/n. 
Proof. That M is commutative with any given transforma- 
tion r of G is seen as follows. We have 

r-wr=r-i5ir-fr-i52r+ . . . =5i+52+ . . . =m, 

since T'^^iT, . . . , r'^^mr are the transformations .Si, . . . , 
Sm over again in some order. Hence MT = TM. Again, 
that M has the form of a similarity-transformation can be 
proved in several ways. We shall here give a very simple proof 
based upon the proposition, following from Theorem 21 : the w- 
elements of each of the matrices of the transformations of G 
do not satisfy a linear homogeneous equation whose coefficients 
are the same for every transformation. Now, the condition 
MT = TM is readily found to imply just such an equation, 
unless M is in the form of a similarity-transformation (a, a, 
...,a). 

Finally, to find the value of a we observe from the formation 
of M that the sum of the elements of its principal diagonal, 
na, equals the sum of the characteristics of ^i, . . . , Sm- 
But these are all equal ( 89); hence a = wx('S'i). 

Lemma 2. If Mi, . . . , Mn are the matrices representing 
each the sum of the matrices of a conjugate set of G, there being 
h such sets, then 

MMt = Cu\Mi-\-c^2M2-\- . . . +CstnMn {s,t = \,2, . . .,h), 

where the coefficients Cai, . . . are positive integers or zero. 
Proof. Let 

M.=5i-f52 . . . -h5, M,=Ri-{-R2-h . . . +R 

then the gsgt matrices in the product MsMt = ljSxR^ must make 
up one or more conjugate sets, since 

T-'{M,M,)T = iT-'M,T){T--'MtT)=M,Mu 



1351 PRODUCT OF CHARACTEKISTIC8 267 

Accordingly, this product is the sum of one or more of the 
matrices M\, . . . , Mu, possibly repeated a certain number 
of times. Hence the lemma. 

136. Theorem 20. Let the number of transformations in 
the different conjugate sets of a transitive group G in n variables 
be g\, g2, . . . , gh, and let the corresponding characteristics be 
denoted by xi, X2, . . , Xa (cf. 89). Then 

(') (f^)(^)=i:-fv) (^. '=>. 2. . *). 

where Ca\, . represent certain positive integers or zero. 

Proof. We substitute in the equation of Lemma 2 the 
canonical forms of the matrices Afi, . . . , Mu as given by 
Lemma 1, and obtain the equation (/3, . . . , /3) = (7, . . . 7), 
where /3 has for value the left-hand member of (1), and 7 the 
right-hand member. 

EXERCISES 

1. Selecting the h equations (1) obtained by keeping 5 fixed while taking 
t=\,2, . . . , h, prove that gsxs/n is an algebraic integer (cf. 116, 7). 

2. Prove that if Ss and Sr^ are conjugate, then 

k t 

i= 1 >= 1 ^' 

where the summation extends over a set of non-equivalent groups into 
which the regular substitution group H breaks up; if 5, and St~ ' are not 
conjugate, the first sum vanishes. 

(Prove that 2^*iXi>^ = if Sv is not the identity; and that if it is, the 
sum equals g. Prove also that if 5s and Sr ' are conjugate, gs=gt, and in 
the right-hand member of (1) we shall then have cai = gs- We assume 
nj=xi^ to be the characteristic of the identity.) 

Corollary 1. The quantity gx equals the product of n by 
the sum of a finite number of roots of unity. 

This follows from the statement of Exercise (1) and 116, 
7. 

Corollary 2. The number of variables n of a transitive 
linear group G is a factor of the order g. 

Proof. The equation from Theorem 18 may be written 

(2) g =1:1X1X1+^2X2X2+ . . . +gX*XA. 



268 CiKUUl' CHAKACTEKI8T1C8 [(Ji.Xiil 

Now, since the sums and products of algebraic integers are 
again algebraic integers, the quantity 

s.=jiixT+^^3ei+ . . . +^r 

n n n n 

is an algebraic integer. It follows that g/ti, being a rational 
number, must be an ordinary integer. 

EXERCISE 

3. Prove that if a transitive linear group G of order g in n variables 
contains a subgroup of order / comf)osed of similarity-transformations, 
then g is divisible by/n (Schur). 

(Prove first that if x* does not vanish, there will be / distinct conju- 
gate sets for which the products gsX^Xs in (2) have the same value.) 

Corollary 3. If a transitive linear group G in n variables 
contains two characteristics Xs, Xi suck that the sum of the w^ roots 
in the product XsXt cannot be written as a sum in which primitive 
roots of index k are absent, theft there is in G a characteristic con- 
taining roots of index k and therefore a transformation whose 
order is k or a multiple of k* 

This follows from the equation (1). By the conditions 
of the corollary, at least one of the characteristics Xc of the 
right-hand member must contain roots of index k. There is, 
therefore, a transformation whose order is divisible by k. For, 
the order w of a transformation 5 = (a, /3, . . .) is the least 
common multiple of the indices of the roots , /3, . . . , since 
5- = (l,l, . . . ) = (a,r, . . . ). 

To illustrate, let Xa= 1+i+i and Xta+a+c^, where 
t = V 1 and a is a primitive fifth root. Here xXt, or 
Aia-\-2ic? 2ao?, cannot be written as a sum which is free- 
from roots of index 20 (fa, etc.) by Kronecker's theorem ( IIC). 

EXERCISE 

4. Prove that if a group in n variables contains transformations of 
orders p and q, two difTerent prime numbers both greater than + l, then 
the group contains a transformation of otder pq. 

136. Theorem 21. Let G = {Si, 52, ... , Sg) be a transi- 
tive linear group in n variables. Then the n^ elements in tlie 

* Bumside, Theory of Croups, second edition, p, 347. 



136) 



GROUP-MATRIX 



269 



matrix yiSi-i-y2S2-\- . . -^-yoSt ( 133), considered as func- 
tions of the g independent variables yi, . . , >*, are linearly 
independent. 

Proof. 1. When a square matrix M of n^ elements is 
transformed into a similar matrix M' by means of a linear 
transformation T in n variables: 

T-'MT = M', M = TM'T-\ 

then the elements of M' are linear functions of the elements 
of M, and vice versa; the coefficients being functions of the 
elements of T. Hence, if the elements of Af are linear functions 
of certain independent variables, the elements of Af' will like- 
wise be linear functions of these variables; and, if among the 
former just / are found to be linearly independent, the same 
will be the case with the elements of M'. 

2. Now consider a regular substitution group H of order 
g written in the form of a linear group. As an example we take 
the symmetric group on 3 letters, which may be written as a 
regular group on 6 letters xi, . . . , xc as follows: 

Si = the identity, S2 = (:*;iX2iC3) (:r4a;5^6) , 



53 = (iCiX3af2) (xiXeXs), 

S5 = (iCiXs) (0:2X4) {X3X6) , 

The matrix yi5i+y252+ . 



54 = ixiX4,) fe^e) (iCaXs), 
56 = (xiXe) (a:2:r5) ix3X4). 
. +3'65'6 is here 



M = 



yi y2 y-s >'4 ys ye 

ys yi y2 ys ya 3'4 

y2 ys yi ye 3'4 >'5 

3'4 ys ye yi y'2 ya 

ys- ya y4 y3 yi y2 



, yo >'4 ys y2 y-s y\ 
New variables may be introduced in ^ so that this group takes 
the intransitive form, namely 

Zl=Xi-\-X2+X3-\-X4-\-X5-\-X6, Z2=X\+X2-\-X3X4Xs Xli, 

Zz=Xi-\-uX2-h 0x^X3, Zi=X4-\-orX5 + <j}Xa, 

Z5=X4-\-uX5-\-ur^XQ, ZQ=Xi-\-urX2-\-OiX3, 



270 



GROUP CHARACTERISTICS 



(Ch. XIII 






B 




















c 


D 














E 


F 




















C 


D 














E 


F 



where w is a primitive cube root of unity. When H is written 
in the new form (we shall denote this by H', and its transforma- 
tions by 5'i, . . . , S'q), the matrix M will become 

A 0' 



M' = 



where 

A=y\+y2-\-yi-\-y^-\-y5+yQ, 

B=yi-{-y2-\-yi-y4.-y5-yQ, 
C=3'i+w2y2+w>'3, D=y^ + o)y5-\-(^yQ, 

E = y^-\-o)'^y5+(ayQ, F = yi-\-oiy2-\-o)^y^. 

Now, there is a transformation, T say, which transforms H 
into fl^' ( 88) : T~^SiT=S'\, Qtc. Therefore, since 

r-(yi5'i+ . . . +yoSg)T=yiS'i+ . . . +y^' 

we have T~^MT = M'. Hence, by 1, there are as many 
linearly independent functions of yi, . . . , y^ among the ele- 
ments A, B, C, D, E, F of M' as among the elements of M, 
namely 6. It follows that A, . . . , F are all linearly inde- 
pendent. 

In the general case, H' breaks up into k non-equivalent 
groups Gi, . . . , Gt, with G, repeated as an equivalent group 
fit times, namely the number of its variables ( 131, Cor. 2). 
Correspondingly, the , component matrices involved in M' 
will have the same elements. Hence, there will be at most 
as many independent elements in M' as are found in a set of 
non-equivalent groups, namely ni^-\-n2^-\- . . . -\-ng^. But 
this number is equal to g ( 132, Ex. 3), the number of inde- 
pendent elements (yi, . . . , yo) in M*. Hence, by 1, all 

* Each element of M will contain a single variable yj. This is proved easily 
from the facts that ^ is a transitive substitution group and every one of its sub- 
stitutions (except the identity) displaces all of the g letters. 



1371 NUMBER OF NON-EQUIVALENT GROUPS 271 

these nr^-\- . . . -\-nt^ elements are linearly independent, and 
the theorem is proved. 

EXERCISES 

1. Prove that the n' elements of each of the matrices of the trans- 
formations of G do not satisfy a linear homogeneous equation whose 
coefficients are the same for every transformation (Burnside). 

2. Prove that if a certain element Cst vanishes in every transformation 
^= [cafi] of a group G, the subscripts s, t being given, then G is not transitive 
(Maschke). 

137. Theorem 22. The number (k) of non-equivalent transitive 
linear groups into which the regular substitution group H breaks 
up (cj. 131) is equal to the total number of sets of conjugate 
substitutions of H. 

The proof follows that of Theorem 21 closely, after we have 
first made equal to each other those of the variables y\, . . . ,yg 
which are factors of conjugate transformations in the matrix 
M. If, therefore, G contains h conjugate sets of respectively 
gi, . . ' , gk transformations, we shall have // independent 
variables, say vi, . . . , Vt,. 

The matrix M' now has the form of a transformation in 
canonical form. Thus, the matrix M' in the example given 
in 136, 2, becomes M' = {A, B, C, C, C, C), where 

A =Vi-\-2V2 + SV3, B=Vi-\-2V2 SV3, C = ViV2. 

In general, let Gj be any one of the groups into which H breaks 
up, and xi, . . . , Xa the characteristics of the various con- 
jugate sets of Gj. Then it follows by Lemma 1, 134, that, 
as far as the variables of Gj are concerned, M' will appear in 
the form of a similarity-transformation (/3/, /3/, . . . , /3;), 
where 

0i = (giViXi-\-g2V2X2-h . . . +ghVhXH)/n. 

If Gj and G'j are equivalent groups, /3/ = /3'^; and conversely 
( 131). Hence, if Gj and G'j are non-equivalent, fijj^fi'j. 

Accordingly, among the g multipliers of M' in its new form, 
there will be just k that are distinct, and these can certainly 
not furnish more than k expressions linearly independent in 
Vif . . . , Vn. On the other hand, the matrix M will contain 



272 GROUP CHARACTERISTICS (Ch. XIII 

just /; linearly independent elements, namely vi, . . . , Vh. 
Hence by 1, 13G, k^Ji, and h of the multipliers, j8i, . . . , /?* 
are linearly independent, say /3i, ^2, , /3*. These h 
expressions can therefore not all vanish unless t>i = zj2 = . . . 

However, if k>h, the expressions /3i, . . . , /3/, must all 
vanish if for i;i, . . . , Vn we put, respectively, the conjugate- 
imaginaries of the characteristics Xi, . . . , x* of the group 
Gh+i (131, Cor. 1). But these quantities, xT, . . . , Xa, 
are not all zero (one of them represents the number of vari- 
ables of Gh+i). We conclude that k>h. Hence, finally, 
k = h. 

EXERCISE 

Prove that if 5i, . . . , Sg are the substitutions of the regular sub- 
stitution group H, and x<^'), x/^^)^ _ _ _ ^ ^^{k) ^hg characteristics of the 
transformations corresponding to St in a set of non-equivalent groups 
into which H breaks up, then these characteristics do not satisfy an equa- 
tion 

0,x/'>+02X*(>+ . . . +CiX|(*> = 

where the coefficients ai, . . . , at are the same for all the g subscripts /. 

138. Theorem 23. No simple group can be of order p'^c^, 
p and q being different prime numbers * 

The proof is divided into two parts: 1. li H is the regular 
substitution group simply isomorphic with a group of order 
g = p^q^, assumed simple, then one of the non-equivalent transi- 
tive linear groups G\, . . . , Gt into which H breaks up ( 131) 
contains (f variables, and one of the conjugate sets of 11 con- 
tains ^ transformations. 2. Under these conditions an 
impossible equation is obtained. 

1. The relation g = p<'q^ = ni^+n2^+ . . . +nt^ (132, Ex. 
3) with the condition that the numbers i, . . . , t are all 
factors of g ( 135, Cor. 2) and that only one of them is unity 
( 132, Ex. 2) implies that at least one of them is greater than 
unity and prime to />; say nt = q">\. 

Again, the relation ^ = /'g^ = gi+g2+ . -\-gh with the 
condition that the numbers g\, - , gn are factors of g and 

* Bumside, Proceedings of the London Mathematical Society, 1904, p. 388. 



138) GROUPS OF ORDER ?>Y 273 

that only one of them is unity (as otherwise U would not be 
simple; cf. 24) implies in the same manner that one of them 
is a power of />; say gs = P^> 1. 

2. We now have a transitive group Gt in ni = (f variables, 
and a conjugate set oi gs = p^ transformations. Let 5^" denote 
one of the transformations of this set, x^*^ >ts characteristic, 
and T the corresponding transformation (substitution) of II. 
We have ( 135, Cor. 1) 

where N represents the sum of a finite number of roots of unity. 
It follows that x^'' =^^', A" being such a sum also ( 116, 7). 
But, x^*^ is already the sum of ^ roots of unity. Hence, either 
all these roots are alike, or x^'^ = 0- The first supposition 
makes S^^ a similarity-transformation, which would be self- 
conjugate in Gt. This being impossible for a simple group, 
we infer that x^'^=0; and this not only for the group Gt, but 
also for every one among the groups Gi, . . . , G* (and their 
equivalent groups), the number of whose variables, like<, does 
not contain /> as a factor. Hence, the sum of the characteristics 
of the transformations corresf>onding to T, from all these groups, 
is of the form 

ix">+ . . . +n*x< = l+/>iV''+gx'+ . . . =1+PN", 

when account is taken of the fact that one of the numbers iti, 
...,*, say i, is unity, and that the corresponding character- 
istic x"^ = 1 

However, this sum is equal to the characteristic x(7') in /7, 
being equal to the sum of the elements of the principal diagonal 
of the matrix of T. Hence, since x(7 ) =0, 

But such an equation is impossible by Kronecker's theorem 

( 116, 6). 

EXERCISE 

Prove that a gioup in which the nunil)er of operators in a conjugate 
set is the power of a prime niuuber is not simple (Bumside). 



274 GROUP CHARACTERISTICS [Oh. XIIJ 

139. Theorem 24. A transitive substitution group of degree 
n and class n 1 contains an invariant subgroup of degree and 
class n* 

Proof. 1. Let g be the order of G, and g' =g/n the order 
of that subgroup G' of G whose substitutions leave fixed a 
given letter. Again, let H represent the regular substitution 
group simply isomorphic with G, and W the regular substitu- 
tion group simply isomorphic with G'. The groups H and 
H' are of degrees g and g' respectively. 

Now let H be resolved into its component linear groups, 
from which we select a set of non-equivalent groups Hi, H2, 
. . . , Ht, the number of whose variables are respectively 
Wi = l, W2, . . . , wjfc. Similarly, let H' be resolved into its 
different linear groups, from which we select a set of non- 
equivalent groups E'l, H'2, . . , H'l, in respectively n\ = \, 
n'2, . . . , n'l variables. The latter set and their equivalent 
groups are all irreducible components of that subgroup of // 
which corresponds to H'; they are, in fact, contained as irre- 
ducible components in the subgroups ol Hi, . . . , Ex (and 
their equivalent groups) which correspond to G' of G. Let us 
suppose, for any subscript s^k, that the subgroup of H, which 
corresponds to G' of G breaks up into/,! groups equivalent to 
H'l, fs2 groups equivalent to H'2, etc. This division may be 
exhibited clearly by the following equation: 

1^. \=UH'i-{-fs2H'2-\- . . . -\-fsiH\, 

Evidently, | Hi \ =H'i, so that 

/ii = l, /l2 = 0, . . . ,/l, = 0. 

Moreover, ( 132, Ex. 3), 

ni2+22+ . . . +Wfc2=g, n'i2+V+ . . . +?=g'. 

Again, if ri( = the identity), T2, . . . , Tg> are the trans- 
formations of \H,\y and Xi( = ), X2*, . . , X/, their char- 

Frobenius, Sitzungsherichte, etc., 1901, pp. 1223-1225. 



139] GROUPS OF DEGREE n AND CLASS n-1 275 

acteristics, while 0ie( = '), 02; . . , dg>, are the corre- 
sponding characteristics of H',, we have {dn = 1) : 

(3) Xt,=fsi-{-Uet2+ . . . +fada. 

2. Assuming for the present that/ii = for a certain sub- 
script s, the theorem is easily proved. If we denote by Sx 
the sum of the n characteristics of Us corresponding to the 
identity and to the substitutions of G of class n,* we have, 
applying Cor. 2, 128, 

=i 1-1 ' 

= y]x-nns, 



or 



(4) 2)^ = 



Ws. 



Hence, there being w characteristics in Sx, and each being 
the sum of roots of unity, the equation (4) can be true only 
if each characteristic x is fis. But then the corresponding trans- 
formation of Hs must be the identity (1, 1, . . . , 1); and 
all such transformations correspond to an invariant subgroup 
of G (cf. 132). This subgroup includes all the substitutions 
of class n and possibly some more (though not all of G, by 
virtue of the condition /,i=0). If it includes more, then this 
new group, of order <g, may be chosen instead of G, and thus 
we would, by a proof by induction, ultimately show the exist- 
ence in G of a subgroup of degree and class n, which would 
obviously be an invariant subgroup of the original group G. 

There are n 1 substitutions in G which permute all the letters; n(jfl) 
which permute all but one; and one (identity) which leaves them all fixed. 

t No two different subgroups of G of class n 1 can have a substitution in 
common. 



276 GROUP CHARACTERISTICS [Ch. XIII 

3. Going back to 1, it remains for us to prove that at least 
one of the numbers /21, /ai, . . , /ti is zero. To accom- 
plish this we start with the following equations (which we shall 
establish in 4) : 

/2i2+/3i2+ . . . +v=i:^' 

(5) /21/22+/31/32+ . . . +/*i/*2 = ^^'a, 



,/ 



g^ 

from which we derive 

Now, since the terms in the left-hand member are positive 
integers or zero, it follows that one of them is unity and the 
rest zero; say 

/22=w'2/21l, /32=W2/31, , fk2=n ifkU 

Substituting in the second of the equations (5) and sub- 
tracting n'2 times the first gives us /2i=0, what we set out 
to prove. 

4. To prove (5), consider the sum 

M = XiXii0u+X,iX2l02+ . . . -^Xv\Xg'\Ba't 

4-X,2Xl201t + Xr2X2202l+ +Xp2Xa'20/i 



+ XrtXlk01t + X,iX202<+ . . . -\-Xf>kXg'i,dg't (K^^^O- 

Let there be h' operators conjugate to T, in G', and there will 
be h'g/g' conjugates to T in G. Then, adding by columns, 
we get (135, Ex. 2) 

Af = /l'(X,iX.l+X,2X,2+ . +Xr*Xrt)^rt=^'^- 



1391 GROUPS OF DEGREE n AND CLASS n-1 277 

Again, substituting for Xu from (3) and adding by lines we 
have (131, Cor. 1): 

t 
r-1 

Hence, equating the two values of M obtained we get 

(6) {)=^frlfn + e,2^fr2fn+ +e^{-\ + ^fn^) 

+ . . . -\-d^'^frifn=Ai + e,2A2+ . . . +M, 

say, where the summation extends from r = 1 to r = ^, and 1 <vS.g'. 
We can now prove the following equation: 

(7) 0=Ai-q+e,2{A2-qn'2)+ . . . +dMi-qn'i) (l^^g')> 
where 

?=^''''.- 

When v>\ the equation reduces to the following by'means of (6) : 
O=-g(l+'20,2+ . . . +n\eu), 

which is true, since the quantity in the parenthesis is the char- 
acteristic corresponding to T, in the regular group H' and is 
therefore zero. If i; = l,the right-hand member of (7) becomes 
(cf. 1) 

Ai-q-[-n'2{A2-qn'2)+ . . -\-n'i{Ai-qn'i) 

=Ai+n'2A2-\- . . . +'^z-5(l+V+ . . . +7) 

I 

= 7^ frtifrl +wVr2+ .. ^-n'tfrlj-n't-qg' 



r-1 



(8) =^frtnr-j-,n't. 



Now, 2^Ii/rr is the number of times the group //'/ (or 
equivalent groups) enters as a component of the subgroup ] H | 
of H. In this group, G' will evidently be represented as an 



278 GROUP CHARACTERISTICS [Ch. Alll 

intransitive substitution group made up of g/g' sets of intransi- 
tivity, of g' letters each. For each such set there are n't groups 
H't. Accordingly, SI^lJ/rtWr = w'tg/g'. Substituting in (8), the 
quantity vanishes. 

Having thus proved (7) for Z' = l, 2, . . . , g', we may apply 
the proposition stated in the exercise, 137, from which it 
follows that every coeflScient Apqn'p must vanish. Equating 
to zero the coefficients for /> = 1 and p = 2, t = l and i = 2, the 
equations (5) are finally obtained. 



PART III* 
APPLICATIONS OF FINITE GROUPS 



CHAPTER XIV 

THE GROUP OF AN ALGEBRAIC EQUATION FOR A GIVEN 

DOMAIN 

140. Introduction. The theory of substitutions and groups 
of substitutions grew out of the investigations by Lagrange, 
Ruffini and Abel of the question of the solvability by radicals 
of the general algebraic equation of degree n. We shall answer 
this question by means of the theory of Galois, which is appli- 
cable to any algebraic equation, whether its coefficients are con- 
stants or depend upon one or more variables. In the latter 
case we must first give a definition of the roots of the equation 
and the concept equality of two functions of the roots. Con- 
sequently, we shall begin with the more concrete, and yet typi- 
cal, case of numerical equations. 

With a given equation we shall associate a certain group of 
substitutions on its roots and shall prove that the equation 
is solvable by radicals if and only if the group is solvable, i.e., 
if each of the factors of composition of the group is a prime 
number. If we regard as known not merely the coefficients 
of the given equation, but also certain constants, such as roots 
of imity, the solution of our equation may be thereby simpli- 
fied and the group altered. In fact, most of our concepts, 
such as the irreducibility of the equation, its group, etc., depend 
upon the constants regarded as known. In order to specify 

*This part was written by L. E. Dickson. 
279 



280 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV 

these constants briefly and clearly, we shall define and employ 
the concept " domain." 

After developing the essential principles of Galois' theory 
of algebraic equations, we shall apply the theory to various 
problems in geometry; first to constructions by ruler and com- 
passes, including the proof of the impossibility of certain con- 
structions of intrinsic and historic interest; then to the inflexion 
points on a plane cubic curve, the 27 straight Unes on a cubic 
surface, and the 28 bitangents to a quartic curve; finally, to 
a general series of problems on contacts of curves. 

141. Number Domains. The set of all rational functions 
with rational coefficients of the complex (real or imaginary) 
numbers ^i, k2, . . - , km is called a domain and denoted by 
R{k\, . . . , kn^- Hence if we perform any one of the four 
rational operations (addition, subtraction, multiplication, 
divison by a number not zero) upon any two equal or distinct 
numbers of the domain, we obtain a number of the domain. 

We assume that each ki is not zero. The domain contains 
every rational numbers, since it contains rk\/ki. The domain 
RiX) is the set of all rational numbers. 

EXERCISES 

1. Every number of R{i), where 1"^= 1, can be given the form a-\-hi, 
where a and b are rational. Every number of /?(V3) is of the form a -\-bV^. 

2. If = -^+V^, then /?() = /?(V^). 

3. If =i-|->/2, R{h V2) = /?({). Hint: i-V2=-3A. 

4. If k is the real cube root of 2, every number of R{k) can be given 
the form a-\-bk-\-ck^, where a, b, c are rational. 

142. Reducibility and Irreducibility. An integral rational 
function /(a;) of degree of a variable x whose coefficiehts belong 
to the domain R is said to be reducible in R if it can be expressed 
as a product of integral rational functions of x each of degree 
<n with coefficients in R; irreducible in R if no such factoriza- 
tion is possible. 

Example 1. The function *+! is reducible in R{i) since it has the 
factors xi, but is irreducible in /?(1) and in R{V2). 

Example 2. x* + l is reducible in any domain which contains either 
y/i, or V^, or i=Vl, or t=il-{-i)/'s/2, but is irreducible in all other 



143] REDUCIBLE AND IRREDUCIBLE FUNCTIONS 281 

domains. Indeed, its linear factors arc r, .r~'; while every quad- 
ratic factor is of one of the forms x'i, x^+axl (*= 2). 

Example 3. x' 2 is irreducible in ^(1). For, if it were reducible, 
it would have a linear factor xa/b, where a and b are relatively prime 
integers, of which b may be taken to be positive. Then a' 26' = 0. If 
b has a prime factor 0{ii> 1), then /3 divides o* and hence divides a, whereas 
a and b have no common factor /3. Thus b=l, o*=2. Hence the positive 
integer a divides 2, so that a=\ or 2, and a'= 1 or 8, whereas a= 2. Hence 
x 2 is irreducible in R(l). 

li f{x) is reducible in R,f{x)=0 is called a reducible equa- 
tion in R; in the contrary case, an irreducible equation in R. 

143. Irreducible Binomials. // p is a prime number and 
if A is a number of a domain R, but A is not the pth power of a 
number of R, then x^A is irreducible in R. 

The roots oi x'' = A may be denoted by 

r, or, coV, . . . , co^'^r, 

where is an imaginary pth. root of unity. Let there be a factor 
with coefl5cients in R oi x^A. It has a constant term of 
the form zLosY, where 0<t<p. By the theory of numbers, 
there exist integers b, c such that btcp = \. Hence R contains 

(wV)* = w*V^+^ = oi'^rA'^r'A', 

where r' is one of the above roots. Thus r' is in R, so that A 
is the pth power of a number r' in i?, contrary to hypothesis. 

144. Theorem. Let the coefficients of the polynomials f(x) 
and g{x) be numbers of a domain R and let f(x) be irreducible 
in R. If one root a of f(x)=0 satisfies g{x)=0, every root of 
f(x) =0 satisfies g{x)=0 and f{x) is a divisor of g{x) in R. 

The greatest common divisor h(x) of f{x) and g{x) is not 
a constant, since it has the factor xa. The usual process for 
finding h(x) involves only rational operations; hence its coef- 
ficients are numbers of the domain R. Since f{x) is irreducible 
in Rf its divisor h{x), with coefhcients in R, is of the same 
degree as f{x), and hence equals cf(x), where c is a number in 
R. But h{x) divides g(x). Hence /(x) divides g(x) in R. 



282 GROUP OF AN ALGEBUAIC EQUATION (Cu. XIV 

EXERCISES 

1. If x'+cx^+dx+c=0, where r, d and e are integers, has a rational 
root, that root is an integer. Hint: Let x=a/b, where a and b are relatively 
prime integers, and multiply the equation obtained from the cubic equation 
hyb*. 

2. An integral root of the equation in Ex. 1 is a divisor of e. Hint: 
It divides x^, cx^ and dx. 

3. x' 3x4-1 is irreducible in R{i). 

4. X* 7x+7 is irreducible in R(l). 

5. State and prove for equations of degree n the theorems correspond- 
ing to those of Exs. 1, 2 for = 3. 

6. x<+x'+x2+x+l = is irreducible in R(\). Hints: It has no 
rational root (Ex. 5). If it has the factors x-'\-ax+r,^x^+bx+r~ ^, where 
o, b, r are rational, then 

0+6=1, o6+r+r-=l, ar-^+br=l. 

Either a = ^{lVE), 6 = K1tV5), r=l; 

r , I 

or o= , b= , r*+r^+r^+r-\-l = 0. 

r+1 r + l 

145. Functions with n\ values. Let R he a. given domain 
which contains all of the coefficients of a given numerical 
equation 

(1) /(a;)^x-Cix"-^+C2:c-*- . . . +(-l)% = 0, 

which, without * real loss of generality, will be assumed to have 
the distinct roots xi, . . . , ocn- There exist integers wi, . . . , 
Wn such that 

Vi = miXi-\-m2X2+ . . . +mrtXn 

gives rise to nl numerically distinct functions Vs when the 
! substitutions s on xi, . . . , Xn are applied to it. For, if 
5 and s' are different substitutions, F, and Vs- are not equal 
/\ , identically as to mi, ... , nin. We can, however, choose 
integers Wi, . . . , Wn which satisfy no one of the !(! 1)/2 
equations of the form Vs = Vs'. In fact, mi=tn2 is the only 
one of these equations involving only mi and m2. Give to 
nil any integral value (say 0) and to W2 any integral value 

* For, if it has a multiple root, fix) and its derivative fix) have a greatest 
common divisor gix) with coefficients in R. Then f{x)/gix) has its coefficients 
in /?, has no multiple root, and vanishes for each root of /(x)=0. 



1461 GALOISIAN RESOLVENTS 283 

9^mi (say 1). Consider the equations involving ma, but not 
nu (J>3); they determine certain values of wa; give to mz 
any integral value distinct from the latter. Next we give 
to Mi an integral value distinct from the values of W4 deter- 
mined by the equations involving m^, but not mi(i> 4) ; etc. 

For mi=0, mj= 1, the values of mt to be avoided are * 

1 1 x-1 X 
0, 1, X, 1-X, -, , ^ , ^_^, 

where 

Xi Xj 

Xl X3 

Thus Xi+mxi has six distinct values under the six substitutions 

1, a=(xiXi), b=(xiX3), c={x2X3), d={,XiX2X3), e={xiXiXj), 

if m is an integer distinct from the above eight numbers. 
We shall often employ as an example the equation 

(2) x+x^+x+l = 0, 

with the roots x,= 1, X2=i=V-l, X3= i. Here X= i, so that xj+wx, 
is six-valued if iht^O, 1, j, \.i, \{\zht), and hence if m=l. The six 
distinct values of x^Xi are 

Fi=X2 xi=l+i, Vi,=XiX3=2i, Vc = XiXi=\i, 
Va=-V Vi=-V, Ve=-Vc. 

146. Galoisian Resolvents. The substitutions on a;i, . . . , 
Xn will be denoted by Si, . . . , 5!, where ^i is the identity. 
If SjSt = Si, and we apply Sj to Vi and then St to the resulting 
function F, , we get Vs^. When k is fixed, buty takes the values 
1, 2, . . . , !, then / takes the same values in some new order. 
Hence St merely permutes 

V,, F,,, . . . , V. 

amongst themselves. Thus the elementary symmetric func- 
tions of these F's are symmetric functions oi xi, . . . , Xn, 

If one of the cross-ratios of four points is X, the others are 1 X, etc. The 
six transformations X'=X, X'=l-X, .... X'=X/(X-1) form a group. Cf. 
Ex. 3, 6 and Ex. 7, 87. 



284 GROUP OF AN AI.(JKRHAFC EQUATION [Ch. XIV 

and hence * arc integral rational functions of wi, . . . , m, 
Ci . . . , Cn with integral coeflicicnts, and therefore are num- 
bers of the domain R. Thus the coefiicients of the polynomial 
in V, given by the expansion of 

(3) F(V)^iV-VS{V-Vs) . . . (V-VsJ, 

are numbers of the domain R. 

If F{V) is reducible in R, let G(V) be that irreducible 
factor in R for which G(Fi)=0. If F{V) is irreducible in R, 
take G{V) to be F{V) itself. In either case, G(F)=0 is an 
irreducible equation in R, having the root Vi; it is called a 
Galoisian resolvent of equation (l) for the domain R. 

The corresponding resolvent for equation (2) in R{\) is 

G{V)^{V-V,){V-Vc) = V^-2V-\-2 = Q. 

For the domain R{i), the resolvent is F Fi = 0. 

147. Theorem. Let <t>{xi, . . . , Xn) be any rational integral 
function, with coefficients in a domain R, of the roots of an equa- 
tion with coefficients in R. Let s he any substitution on the roots 
and let it replace <i> by 4>s, and an nl-valued linear function Fi, 
with coefficients in R, by F,. Then 



(4) 0,= 



F'iVsY 



where X is a polynomial with coefficients in R, while F' is the 
derivative of the polynomial (3) with coefficients in R, so that 
F'(Vs) 7^0. Thus </)s is the same rational function p(Va) of V, 
that <t> = <t>i is of V\: 

If S}St = si, then Sk replaces <^s^ by </),,j. Thus St permutes 
<^i, . . . , <^,^, in the same manner that it permutes Fi, . . . , 
F,^, ( 146). Hence the terms of 

(r,\ xm-^ ^^^^^ +^ _W). , . . F{V) 

(.3) Mn=<^i^73y^+<^.',prrir+ -^K.v-v.^, 

* Several detailed proofs of this fundamental theorem on symmetric functions, 
which is frequently applied below, are given in Dickson's Eicmcntary Theory of 
Equations. 



147] GALOISIAN RESOLVENTS 286 

are merely permuted amongst themselves by any substitu- 
tion on a:i, . . . , Xn. Thus the coefficients of the polynomial 
X(F) are rational integral symmetric functions oi xi, . . . , Xn 
with coefficients in the domain R, and hence equal numbers 
of R. Taking V=V we obtain X(F,) = 0,F'(F,). Since 
F'(F,)?^0, weget (4). 

Example. Recurring to the special equation (2), we shall obtain 
the explicit expressions (4) for the case = X2, Fi=X2 jci. Then 

(3') F(F) = F+4F<+4P^2 + 16, 

(50 x(F)=F(F)|-^^+^^+-^^+-^^+^^+-^^ 
^ '\V-V, F + F. V-Vb V-Vc V+Vi, V+Vc 

= -2F5-4F*-12F3-8F2-16F-48, 

as shown by inserting the values of ^i, . . . , Vc given at the end of 145. 
Hence 

/T. N -2FiS-4Fi*-12Fi3-8Fi2-16Fi-48 

Xt=p{Vi) = . 

6Fi*+16Fi3+8Fi 

In view of the theorem, we have 

Xi=p{Va), X2=p{Vb), X3=p{Vc), X3=p{Va), Xi=p(Ve). 

These results may be verified by evaluating the expressions. . 

The numerator and denominator of the above fraction for X2 may be 
expressed as linear functions of Vi by means of the relation Fi* 2Fi4-2=0 
of 146. We get 

-48Fi+32 _ (-3Fi+2)(Fi+2) _ -.3Fi^-4Fi+4 _ -10Fi + 10 , 
16Fi-64 ~ (Fi-4)(Fi+2) ~ F,2-2Fi-8 ~ -10 ~ ' 

While therefore X2 is numerically equal to Fi 1 (each being t), it is not 
admissible to take this reduced function r(Fi) = Fi 1 as the function 
p(Fi) of the theorem, since it would no longer be true that, by applying 
the substitution a'={xiX^, we would have xi = r{Va)- Indeed, if we apply 
a to Vi \=X2Xi\, we obtain Xi.X2\9^Xi. The explanation is that 
we should reduce the second member of the true relation Xi = p(Fa) by 
means of F*o+2Fa+2=0; we thus obtain 

16Fa+64 

which is the correct value of X\. Since Vc satisfies the same quadratic 
equation as Vi, our first reduction yields also the true relation XtVcl. 



286 GROUP OF AN ALGEBRAIC EQUATION [Cu. XIV 

We therefore see why we obtain a true relation if we apply the substitu- 
tion c to the members of Lhe reduced relation X2= Ki 1, and why it would 
be accidental if we obtained a true relation when we apply to X2=Ki 1 
any substitution other than c and the identity substitution. This example 
brings us to the core of our subject and indicates the care which must be 
taken in its development. 

148. The Group of an Equation for a Domain R. Let the 

roots of a Galoisian resolvent G{V)=0 of the given equation be 

(6) Fi, Va, V,, . . . , F 

in which the subscripts denote the substitutions on :ri, . . . , Xn 
by which these F's are derived from Vi. We shall prove the 
Theorem. The g substitutions 

(7) 1, a,b, . . . ,1 

form a group G, called the group of the given equation (1) for 
the domain R. 

We are to prove that the product rs of any two of the sub- 
stitutions (7) is one of those substitutions. Take F, as the 
function in 147. Then 

where X is a polynomial with coefficients in R. Since F, is a 
rootof G(F)=0, 

X(F) 






= 



'(F); 

is satisfied when F=Fi. Multiplying the left member by the 
gth power of F'{V), we obtain an integral function /(F) of V 
which vanishes for F = Fi and has numbers of R as coefficients. 
Hence the root Fj of the irreducible equation G(F)=0 in R 
is a root of /(F) =0 ( 144). Since F'{Vs)7^0f we may divide 
/(F,) by the gth. power of F'{Vs) and get 



'=K^)=^(^"^- 



Hence Vn is one of the functions (6) and rs one of the substi- 
tutions (7). 



149) PROPERTIES A, B OF THE GROUP 287 

Example. For the domain of rational numbers, a Galoisian resolvent 
of the cubic equation (2) was seen in 140 to have the roots Vi and Ve. 
Hence the group of (2) for R{1) is {l, (a:2a;3)|. But for the domain R(i), 
a resolvent is F Fi = and the group is the identity. 

149. Characteristic Properties of the Group G of a Given 
Equation for a Domain R. Let ^/^ be the quotient of two 
rational integral functions of the roots with coeflScients in R, 
such that ^?^0. We have (4) and 

^' F'iVsY 

where m is a polynomial with coefficients in R, and fi(Vi)9^0. 
If 5 is a substitution of G, then xJ/st^O. For, if )u(F) =0 has the 
root Vs, it has also the root Vi in common with the irreducible 
Galoisian resolvent G{V)=0 ( 144). Hence the functions 

<f>, \{Vs) 



^s niVs) 



is = l,a, ...,/) 



are defined for each substitution s of the group G. 

Suppose that these g functions are equal numerically, in 
other words, that 0/^ is unaltered in value by all of the 
substitutions of G. Then 

</>_l[X(Fi) \{Va) ,X(F01 

The second member is a rational symmetric function, with 
coefficients in R, of the roots (6) of G(F) =0 and hence equals 
a rational function of its coefficients, which belong to R. H[ence 
0/^ equals a number in R. 

A. If a rational function with coefficients in R of the roots 
of an equation with coefficients in R remains unaltered in value 
by all of the substitutions of the group G of the equation for R, 
it equals a number in R. 

B. Conversely, if a rational function of the roots with coef- 
ficients in R equals a number in R, it remains unaltered in value 
by all of the substitutions of G. 

It remains to prove B. Let <t>/4/ equal the number r in R. 
Then \iV)/fi{V)-r vanishes for V=Vi. Hence the equation 



288 GROUP OP AN ALGEBRAIC EQUATION [Ch. XIV 

X(V') f/u(V)=0 with coefficients in R is satisfied by every 
root V, of the irreducible equation G{V) =0 ( 144). Hence 

so that </>/^ is unaltered in value by the substitutions of G. 

Example. Consider a cubic equation, like (2), with a rational root 
Xi and no multiple root. By property B with Xi as the rational function, 
its group for any domain containing the coefficients has no substitution 
other than 1 and (xiXs). If the domain contains Xn and hence also Xj, 
the group is the identity; this is the case with equation (2) for R{i). In 
the contrary case, there must, by property A, be a substitution altering 
Xj, so that the group is |l, (x2X3){. 

Since an !-valued function Vi with coefficients in a given 
domain R can be chosen in an infinitude of ways, there are 
infinitely many Galois resolvents G{V)=0. Our definition 
of the group G of the given equation for the domain was based 
upon a single such resolvent, i.e., upon a particular Fi. It 
is a fundamental proposition that different functions Vi always 
lead to the same group G. This follows from the 

Theorem. The group of a given equation for a given 
domain R is uniquely defined by properties A and B. 

First, suppose that G' = \l, a', b', . . . ^ m'\ is a group 
for which property A holds. Then the coefficients of 

<1>{V)^{V-V{){V-V^){V-V,) . . . (F-F,), 

beiilg symmetric functions of Fi, . . . , V,^, are unaltered 
numerically by the substitutions of G' and hence equal num- 
bers in R. Since the equation 0(F)=O, with coefficients in 
R, admits one root Vi of the irreducible Galoisian resolvent 
G(F)=0, it admits all of the roots (6) of the latter ( 144). 
Hence 1, a, . , . , / occur among the substitutions of G\ so 
that G is a subgroup * of G' . 

Second, suppose that r = {l, a, /3, . . . , x} is a group 
for which property B holds. Then the Galoisian function 

* In Part III, a group is included among its subgroups. 



150) TRANSITIVE GROUP . 289 

G{Vi), being equal to the number zero in R, remains unaltered 
in value by the substitutions a, 0, . . . , x, so that 

= G(Fi)=G(O= . . . =G(F,). 

Hence Va, . . . , V^ occur among the roots (6) of G{V)=0. 
Thus r is a subgroup of G. 

liG' = T, then G' = G. 

In view of its repeated application below, we state our 
second result as the 

Corollary. // every rational function of the roots with 
coefficients in R which equals a quantity in R is unaltered in 
value by every substitution of a group T, then T is a subgroup of 
the group G for R of the equation. 

160. Transitive Group. We shall make much use of the 

Theorem. // an equation is irreducible in a domain R, 
its group for R is transitive, and conversely. 

Consider an equation f{x)=0 irreducible in R. Contrary 
to the theorem, suppose that its group G for R is intransitive 
and contains substitutions replacing Xi by iCi, 0:2, . , Xm, 
but none replacing xi by one of Xm+i, . . . , Xn. Hence 
every substitution of G permutes xi, . . . , Xm amongst them- 
selves and thus leaves unaltered any symmetric function of 
them. Hence 

g{x) = (x-Xi)(x-X2) . . . (X-X,n) 

has its coefficients in R, in view of property A. Thus f(x) 
has the factor g(x) in R, contrary to its irreducibility in R. 

To prove the converse, let G be transitive and the equation 
f{x)=0 be reducible in R. Let the preceding function g(x) 
be a factor of f{x), the Coefficients of g{x) being in R and its 
degree m being less than n. Since g{xi) equals the number 
zero of R, it is unaltered by every substitution of G (property 
B). Since G is transitive, g{xi)=0 for t = l, . . . , . This 
tontradicts m<n. 

Example 1, Find the group G of x 7x-|-7 = for /?(1). 

The equation is irreducible (Ex. 4, 144), so that G is transitive. It 



290 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV 

will therefore be the alternating group G={l, (xix^xt), (xix^i)] if shown 
not to be the symmetric group. The square of the function 

(8) , ^ = {xi -Xi)ixt Xt) (xj - xi) 

is the discriminant * 49 of our cubic, so that \f/ equals a number 7 
of the domain R{1). A transposition changes ^ to ^^ and hence is not 
in G (property B). Hence G^Gz- 

ExABiPLE 2. Find the group G oi x*+l = iov R{1). 

II Xi, . . . , Xi are the roots of x* +ax^ +bx^ +cx+d=0, then 

yi=xiXo-\-X3Xi, y2=xiX3+xtXi, yi=XiXi+XiXt 
are the roots of the resolvent cubic equation 

(9) y^-by^+{ac-id)y-a^d+4bd-ci=0, 

as shown f by finding yi+yi+yz, - , yiy2yi, or by Ferrari's method 
of solving the quartic equation. For x< + l = 0, (9) is y 4y=0 and 
has three distinct rational roots. Hence (by B), each substitution of 
G leaves yi, yt and yz formally unaltered. Now yi is unaltered only by the 
substitutions of the group 

(10) G8=|l, (12), (34), (12)(34), (13)(24), (14)(23), (1324), (1423)}, 
while yz is unaltered only by the substitutions of 

(11) C'8=}1, (13), (24), (13)(24), (12)(34), (14)(23), (1234), (1432)}. 
The substitutions common to these groups form the group 

(12) G,= {1, (12)(34), (13)(24), (14)(23)}. 

By Ex. 2 of 1 142, ac* + l is irreducible in i2(l). Hence G=Gi. 

Example 3. Find the group G of x<+x'+a;2 4-x+l = for /?(1). 

The roots may be denoted by Xi = t, X2=e^, Xi = t*, Xi=e*, where c is 
an imaginary fifth root of unity. Then ^2 = 2, while yi = '+ and 
y=t*+( are the roots i( liVs) of y^+y 1 = 0, as may also be shown 
by use of (9). Thus G is a subgroup of G's, which leaves ya formally unal- 
tered. But the substitutions (13), (24), (12)(34) and (14)(23) replace 
Xj Xi*, whose value is zero, by functions which are not zero. Hence 
G is a subgroup of the cyclic group generated by (1234). Since the equa- 
tion is irreducible in R(l), by Ex. 6, 144, G is this cyclic group. 

*For X* + px-\-q= 0, if'^=-4p*- 27 qK 

t Cf . Dickson's Elementary Theory of Equations, p. 39, 3. 



1501 TRANSITIVE GROUP 291 

EXERCISES 

For the domain of rational numbers, find the group of 

1. x-l = 0. 2. (x-I)(x + l)(:c-2) = 0. 

3. x-9x+9 = [compute (8)]. 4. x-2 = 0. 

5. For the domain /?(&;), where w is an imaginary cube root of tinity, 
the group of x' 2 = is of order 3. [Compute (8)]. 

6. For the domain RU), the group of x< + l = is of order 2. 

7. Find the group G of a reciprocal quartic equation 

X* +ax^ +bx^ +ax +1 = 

for the domain R=R{a, b), when it is irreducible in R. 

Hints: Choose the notation for the roots so that Xiaf2=a;3a:4=l. Then 
one root of the cubic (9) is ^1 = 2; thus y^ and yz are the roots ^blVA 
of y*4-(2-6)3'4-a*-26=0, where A = (^b+iy-a^-. The three y's are 
distinct; for example, yiyi=(xiX4)ix3X2). Hence G is a subgroup 
of Gg, given by (10). Further, G is Gt, given by (12), if and only if Va 
isia R. 

By the usual substitution v=x+l/x, our quartic becomes v*+av+b2 
=0, whose roots are therefore iCi+X2 and X3+x^. Its discriminant 
B=a^4{b2) is thus the square of t=Xi+XiX3Xi. Why is MO? 
Now (1324), (14) (23) and (13) (24) replace /by -t, while the first four 
substitutions in (10) leave / unaltered. Again, ytyz is unaltered only 
by the subgroup G* of Gs, being changed in sign by the remaining four 
substitutions of d. Hence {(yiys) is unaltered only by the subgroup 
Ci generated by (1324). Hence G=C^ if and only if VaB is in R. 

If a transitive subgroup of Gs does not contain (1324) or its inverse 
(1423), it contains (13)(24) and (14)(23), the only remaining substitutions 
replacing Xi by Xj and Xt, respectively, and hence is G*. Thus if G is not 
Ci or G4, it is Gs. It follows by formal logic that G=G8 if and only if 
neither y/A nor VaB is in R. 

8. If the quartic in Ex. 7 is reducible in R, it is the product of two fac- 
tors x*-\-px+r and x^+qx+1/r, where p, q, r are in R, and 

P 1 

p+q=a, -+rq=a, r-^ \-pq=o. 

r r 

If r=l, p and q are in R only when Vb is in R. If r= 1, then a = 
and y/b2 must be in R. If rVl, we may eliminate p and q and 
obtain for y=r+l/r the quadratic * in Ex. 7 with the roots yt, y,; thus R 
must contain V]4 and the square roots of (Jft 1V^)* 4, the latter 
being the values of (r l/r)*=y2 4 for y=yj, yj. By Ex. 7, ABy^O if 
the four roots are distinct. 

9. If a=b=0, then A = l, 5 = 8, and x* + l is irreducible in RiD, and 

* Except for a=0; then P=q0, y=b. 



292 GROUP OF AN ALGEBRAIC EQUATION ICn. XIV 

has the group G4. If u = 6= 1, then .4=5/4. B=5, anda;+x+a;*+x + l = 
is irreducible in /?(1) and has the group d. 

Let /(ac) = aox"+flix"~ '+. . . =0 be an equation with rational coef- 
ficients, irreducible in the domain of rational numbers. Prove * Exs. 
10-14: 

10. If there is a complex root of absolute value unity, the equation 
is a reciprocal equation of even degree. 

11. If there is a root r+si, where r is rational, then n is even and the 
n roots may be paired so that the sum of the two of any pair is 2r, whence 
r= fli/(nflo). In particular, if r=0, the equation involves only even 
powers of x. 

12. If there is an imaginary root a+bi whose norm a^-\-b^ is rational, 
then n is even and the n roots can be paired so that the product of any two 
of a pair is o*+6^ 

13. If there is a root whose absolute value p is rational, p can be 
expressed in terms of the coefficients. (p**=a/oo.) 

14. If we set x=py in the equation in Ex. 13, we obtain a reciprocal 
equation in y. 

Equations whose Coefficients Involve Variables, 151-6 

151. Definition of the Roots. We begin with the so-called general 
equation (1) whose coefficients ci, . . . , Cn are independent complex vari- 
ables. Let a(ci, . . . , Cn) be its discriminant. Let Oi, . . . , an and 
Ai, . . . , ^n be any two sets of constant values of ci, . . . , c for which 
A?^0. We shall prove that the A's can be derived from the o's by con- 
tinuous variation such that, for each intermediate set of values, Ay^O; 
expressed in geometrical language, there is a continuous path from the 
point (a) to the point (A) not passing through a point of the locus A = 0. 

We shall prove this by induction from nl to n, assuming that, if 
P(ci, . . . , Cn-i) is any pKjlynomial in Ci, . . . , Cn-i, zero neither at 
(/i, . . . , In-i) nor at (1, . . . , Ln-i), there is a continuous path from 
(/) to (L) not passing through a point of the locus P=0. Neither of the 
polynomials 

A(a, . . . , C-i, On) A(Ci, . . . ,Cn-u An) 

is identically zero, since the first is not zero when each ' = a<, and the second 
is not zero when each Ci = Ai. Hence there are constants o< for which 

A(ai, . . . , an-i, Un)5^0, A(ai, . . . , a-i, An)7^0. 

Thus there is, by hypothesis, a continuous path from (ci, . . . , On-i, a) 
to (ai, . . . , on-i. On), not passing through a point of P=A(ci, . . . , 
Cm-\, c) = and composed only of points with c = a, and hence not passing 

Exs. 10, 11, 13 are due to Dr. A. J. Kempner. 



1521 FUNCTION DOMAINS, EQUALITY 23 

through a point of A(ci, . . . , Cn) = 0. The equation A(ai om-uv) 

= in r has only a finite number of roots. Hence v can be varied contin- 
uously from On to An SO that A(ai, . . . , an - 1, v) 9^0 at each intermediate 
point. The combination of our two paths gives a continuous path from 
(o) to (ai, . . . ,an-i, An) not passing through a point of A = 0. 
Similarly, there exist constants /3j for which 

A03i, . . . ,j3-:, an-i, An)7^0, A(fii, . . . , /3-,, An-i, ^)70. 

By hypothesis, there is a continuous path frcm (ai, . . . ,_!, An) to 
(/3i, . . . , A,-s, a-i, An), not passing through a point of a(ci, . . . , 
Cn-i, an-i, An) = and composed of points with Cn-i=a_i, Cn=An, and 
hence not passing through a point of A(ci, . . . , <^) = 0. Evidently 
there is a continuous path from our final point to E={0i, . . . , /3_i, 
An -I, An) not passing through a point of A = 0. We now have a continu- 
ous path from (a) to E not passing through a point of A=0. Proceeding 
in this manner, we finally get such a path from (a) to (^4). 

Let Xi", . . . , ocn" he the n distinct roots of the equation with the 
coeflBcients ai, . . . , On- These roots receive increments as small in 
absolute value as we please when Oi, . . . , On are given increments suf- 
ficiently small in absolute value.* Hence if we proceed along our path 
from (a) to (^), we obtain a definite coordination of the roots x'l, . . . , x'n 
of the equation having the coefficients Ai, . . . , An with the initial values 
Xi", . . . , Xb". Thus the latter and definite paths radiating from (a) 
lead to n functions Xi, . . . , x of Ci, . . . , Cn uniquely defined for every 
set of c's for which A^^O, and called the roots of the general equation (1). 
In fact, for a particular set of c's, the roots of the equation are the values 
of the functions Xi, . . . , Xn for those c's. Our main investigation is the 
comparison of a rational function of the roots with that derived by a 
substitution on the roots; hence we shall not be interested in values of the 
c's for which A=0, i.e., for which two or more roots become equal. 

The same scheme defines a fortiori the roots of any equation whose 
coeflScients are functions of one or more variables. We retain only those 
sets (A) which are sets of values of our present coefficients. The fact 
that certain of the sets intermediate to (a) and (^4) are not now values of 
our coeflBcients does not disturb the coordination of the roots at (A) with 
those at (a). The scheme therefore assembles the root values into root 
functions. 

152. Function Domains, Equality, Group of an Equation. 
Instead of a domain composed of constants, we now employ 
a domain R{ki, . . . , i^) composed of all rational functions 

*The roots are continuous functions of the coefficients. For a proof, see 
Weber's Algebra, vol. 1, 1895, p. 132. 



294 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV 

with rational coefficients of ^i, . . . , ^, either all of which 
are given functions, or certain of which are functions and the 
others are constants. For example, -R(V3, k) is composed of 
all rational functions of the variable k with coefficients of the 
form a+ftVs, where a and h are rational numbers. 

Two polynomials and ^ in the variables V\, . . . , Vi 
with coefficients in R are called equal if they have the same 
numerical value for every set of numerical values which v\, 
. . . , vi, ki, . . . , km can assume. For example, if vi and 
V2 are the roots of x'^-\-kx-\-l=0, then t'i+z'2= ^z'i2^2- 

The equality of two rational functions of z^i, . . . , z^i is 
defined similarly, but with restriction to those sets of values 
of the v^s and ^'s for which each denominator is not zero. 

If \{/ is derived from (/> by a substitution s on Vi, . . . , Vi 
and if ^ = <^ in the present sense of equality, we shall say that 
<j> is unaltered by s. 

The definitions and theorems in 142-4 concerning irre- 
ducibility evidently hold for the present generalized domain 
R. Each element (function) in R is conveniently called a quan- 
tity in R. 

Proceeding as in 145-9, we see that an equation whose 
coefficients ci, . . . , c, are any given functions or constants 
has a definite group G for any given domain containing ci, 

. . . , Co- 

153. Group of the General Equation. Let the coefficients 
Ci, . . . , Co be independent complex variables. Let Xi, . . . , 
Xn be the roots in the sense of 151. 

Lemma. // a rational integral function ^{xi, . . . , Xn) 
of the roots with constant coefficients is zero for every set of values 
of c\, . . . , Cn, each coefficient of \[/ is zero. 

We consider those sets of values of ci, C2, . . . , c which 
are the values of the elementary symmetric functions Sz;i, 
2z'i2;2, . . . , Vi . . . Vn of the independent variables vi, 
. . . , Vn- For each set of values of the v's we therefore obtain 
a set of values oi xi, . . . , Xn forming a permutation of the 
v's. Consider the product P of \l/{vi, . . . , ^) by the func- 
tions obtained from it by applying the various substitutions, 



153] GROUP OF THE GENERAL EQUATION 295 

other than identity, on i^i, . . . , ti.. For every set of t>*s, 
one factor of P is a value of \l/(xi, . . . , Xn) and hence is zero. 
Thus P is zero identically in the v's. Hence some factor ^ of 
it is zero identically. 

Theorem. The group of the general equation for the domain 
R defined by its coefficients and any chosen constants is the sym- 
metric group. 

The coefl&cients of its Galoisian resolvent G{V)=0 are 
rational integral functions of ci, . . . , c with constant coef- 
ficients. Replace Ci, . . . , c by the elementary symmetric 
functions of xi, . \ . , Xn. Then G{V) becomes a poly- 
nomial P{V) whose coefficients are rational integral functions 
of the x's with constant coefficients. Let Vi = I,miXt, where 
mi, . . . , mn are distinct integers, be the function used in 
constructing G{V). Then PiVi) is a rational integral func- 
tion of the x^s with constant coefficients which is zero for every 
set of values of the c's. By the Lemma, P{Vi) is zero identi- 
cally in the x's. The function derived from it by applying 
any substitution 5 on the x's is therefore zero identically in 
the x's. Since the coefficients of G{V) are unaltered by this 
substitution, we get G(Vs)=0. Hence every substitution 5 
occurs in the group of the equation. 

EXERCISES 

1. Prove the last theorem by showing that properties A and B in 149 
hold when G is the symmetric group. Note that il 4t=(t>s for all values 
of the c's, then, by the Lemma, (f>=<i>s identically in the x's; if this is true 
for every substitution s, <t> is symmetric and hence is a rational function 
with rational coefficients of Ci, . . . , Cn and the coefficients of <f>. Next, 
if <t>ixi, . . . , Xn) equals a rational function of the c's and hence a rational 
symmetric function ^ of the x's, for every set of c's, then ^s ^ identically 
in the x's (Lemma), so that <A is symmetric, and property B holds. 

2. If Gis the group of /(x, c)=x" cix~^-f- . . . Cn = Ofor the domain 
R=R(ci, . . . , Cn, ki, . . . , ki), where ki, . . . , ki are constants, and 
if c'l, . . . , c'n are values which ci, . . . , c^ can take, then the group 
G' ol fix, c') = for R'=R(c'i, . . . , c'n, ki, . . . , h) is G or a subgroup 
of C. 

Hint: If G(V,,c) = is the Galoisian resolvent of /(x, c) = for R, the 
group of /(x, c') = for R' is G or a subgroup according as C(K, c') is irre- 
ducible or reducible in R'. 



296 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV 

3. Hence the group of the general cubic x*cix*+cixci=0 for 
Rict, Ci, Ci) is Gt, since that of x' 2 = for R{\) is G. 

4. For a and b independent variables, the group of x* +fl* + JaC +ox 
+1=0 is G. 

5. The group of an irreducible quartic equation for a domain/? is the 
symmetric group if no one of the roots yi, y2, yz of the resolvent cubic 
equation (9) is in R, and if the product P of the differences of the y's is 
not in R. 

Hints: Its group G is not (10) or (11), and not the group G"g leaving 
yj unaltered, nor their common subgroup (12), nor ?. cyclic group of order 
four, necessarily contained in one of these groups of order eight. The only 
remaining transitive groups are the alternating and symmetric groups, 
16. But G is not the former in view of P. 

6. The group of an irreducible quartic for R is Gn if that of the resol- 
vent cubic (9) is Gt. 

164. Rational Functions belonging to a Group. Let 1, 
a,b, . . . , ^ be all of the substitutions of the group (j of a given 
equation for a given domain R which leave unaltered (in the 
sense of 152) a given rational function ^{xi, . . . , Xn) oi the 
roots with coefficients in R. Since \p \f/a has the value zero, 
we have xh> ^<a = by property B. Hence \t/ab = ^ and ab 
is in the set 1, a, . . . , ^, which therefore forms a group H. 
We shall say that ^ belongs to the subgroup H of G. 

Example. Let Xi and 'a;2= a;i be two roots of r;<+l = 0. The only 
substitutions of its group (12) for R{1) which leave xr unaltered are 1, 
(12) (34). Hence they form the subgroup to which Xi^ belongs. 

Conversely, let H be any given subgroup of G. Let Vi 
be any !-valued function of the roots with coefficients in R, 
and let Vi, Va, . . . , Vt be the functions obtained from Vi 
by applying the substitutions of H. If p is a suitably chosen 
number in i?, 

^=(p-Fi)(p-F) . . . (p-n) 

is a rational function of the roots Xi with coefficients in R which 
belongs to H. For, if 5 be any substitution of H, then s, as, 
. . . , ks are distinct and are in H, and hence form a permuta- 
tion of 1, a, . . . , ^. Thus ^ equals 

^.=(p-F,)(p-F) . . . (p-FJ. 



154] RATIONAL FUNCTIONS BELONGING TO A GROUP 297 

But, if 5 is a substitution of G not in H, then ^, is not identical 
with ^ as to the variable p, since V, is different from Vi, Va, 
. . . , Vt. We may therefore choose an integer p such that 
4^ belongs to H. This proves 

Theorem 1. Every rational function ^ with coefficients 
in R of the roots of an equation with the group G for the domain 
R belongs to a definite subgroup of G. There exist such functions 
^ belonging to any assigned subgroup of G. 

We next prove the important supplementary 
Theorem 2. // a rational function \f^, with coefficients in a 
domain R, of the roots of an equation with the group G for R, be- 
longs to a subgroup H of index v under G, then the substitutions 
of G replace ^ by exactly v distinct functions; they are the roots 
of an equation 

(13) g{y)^iy-^i){y->p2) . . . (3'-^.) 

with coefficients in R and irreducible in R. 
As in 10, let 

(14) G = H+Hg2+Hg^+ . . . +Hg,. 
Let h be any substitution of H. Then 

^A(r, = ( ^^^)g^ = (rA)*/, = 4^9^- 

Thus \p takes at most v values under G. But, if 

ht = hj U<i), 

then 4^g^g-i = y^^ so that gtgj'^ is a substitution h oi G leaving ^ 
unaltered and hence is in H. Then gi = hgj, contrary to (14). 
Thus 4^1, ^2, . ' . , ^y are distinct, where ^ has been written 
lor ^fff. They are called the conjugates to ^ = ^i under G. 

Any substitution s ol G _ merely permutes ^i, . . . , ^^, 
amongst themselves. For, any product gtS may be written in 
the form hgj where A is in ^; then 

Ms = ^g^ = fhtj = hj = h- 

Hence the coefficients of (13) are unaltered by every sub- 
stitution of G and therefore equal quantities in R. 

If giy] has a factor with coefficients in R, it has a factor 



298 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV 

y{y) which Is zero for y = \J/i and hence (by B of 149) for 
y = ^2, . . . I y = ^,- Thus y=g, so that g{y) is irreducible 
inR. 

Example 1. The group G of x'+x^-\-x+l = for R{1) is {l, (ocxj)}, 
if xi= 1 denotes the real root. The conjugates to \//i = X2xi under G 
are lAi and i2=XiXi; they are the roots of y'* 2y+2 = 0. 

Example 2. The group G of x*+l = for R{i) is {l, (xiXj)(x2iC4)} if 
a:i = , ici=i, xs= , a;4= le, where =(l+i)/'^, so that t''=i. The 
conjugates Xi and xz to xi under G are the roots of y^ i=0, which is irre- 
ducible in R{i), to which e does not belong. 

155. Galois' Generalization of Lagrange's Theorem. // a 

rational function <j), with coefficients in a domain R, of the roots 
of an equation f{x) = 0, with the group G for R, remains unaltered 
by all those substitutions of G which leave unaltered another rational 
function \f/ of the roots with coefficients in R, then (f> equals a rational 
function of rj/ with coefficients in R. 

In case ^ is an !-valued function Vi, the only substitution leaving ^ 
unaltered is the identity, and this leaves any <j) unaltered. For this case, 
the theorem states that any rational function <i> with coefficients in R 
equals a rational function of Vi with coefficients in R. This follows from 
the like result in 147 for the rational integral numerator and denominator 
of <t>. 

Let H be the subgroup of G of index p to which ^ belongs. 
By means of (14), we obtain the v distinct conjugates ^i, . . . , 
^, to ^=^1 under G. Since every substitution h oi H leaves 
unaltered, each product hgt replaces <j) by (}H = (t>g^. Any sub- 
stitution 5 of G replaces ^< by a certain rj/j (end of 154) and 
hkewise <fH by (f>j. Thus, for ^(3;) defined by (13), 

xW-g(y)(-^+-^+ . . . +-i~r) 

\y-tj/i y-}p2 y WvJ 

is an integral function of y each of whose coefficients is unaltered 
by every substitution of G and hence is in R. Taking ^1 = ^ 
as y, we get <^ = X(^)H-g'(^). 

The theorem will be shown to be a generalization of 
Lagrange's Theorem. If a rational function <f) of the 
independent variables x\, . . . , Xn remains unaltered by all 
those substitutions on Xi, . . . , Xn which leave unaltered another 



156] ADJUNCTION TO THE DOMAIN 299 

rational function \f/ of xi, . . . , Xh, then <t> equals a rational 
function of \l/ and the elementary symmetric functions 

Cl=TtXi, . . . , Cn=XiX2 . . . Xn. 

The group of the equation with the coefficients ci, . . . , Cn 
for the domain R, defined by ci, . . . , Cn and given constants 
ki, is the symmetric group. For, property A then states that 
any symmetric function of Xi, . . . , Xn with coefficients 
rational in the ^'s is in R, and is the well-known theorem on 
symmetric functions. Conversely, a function equal to a quan- 
tity in R is symmetric. 

Example. y2=xiX3+a:2:j;4 is unaltered by all of the substitutions 1, 
(13), (24), (13)(24), which leave ^ = Xi+XiXiXi unaltered. We see 
that 

y2=l(i^Ci^-\-'iCi), Ci = 2xi, C2=2^iX2. 

When a rational function of independent variables Xi, . . . , 
Xn is unaltered by each substitution of a group H on the re's, 
but is altered by every substitution not in H, it is said to belong 
to the group H. We need not specify as in 154 that fl^ is a 
subgroup of the group G of the equation with the x's as roots, 
since G is now the total symmetric group. 

156. Effect on the Group by an Adjunction to the Domain. 
Let G be the group of /(x) =0 for a domain i? = i?(^i, . . . , km) 
containing the coefficients. Let R' = R{^, ki, . . . , km) be 
the domain composed of the rational functions oi xj/, ki, . . . , 
km with rational coefficients. This enlarged domain R' is said 
to be derived from R by adjoining the quantity ^. If the 
irreducible Galoisian resolvent G{V)=0 for the initial domain 
R remains irreducible in R\ the group of f{x) = for R' is 
evidently G. But if it reduces in R', let G'{V) be that factor 
of G{V) which has its coefficients in R', is irreducible in R', and 
vanishes for V = Vi. Then if Fi, Fa, . . . , V^ are the roots 
of G'(F)=0, the groupof/(x)=Ofor/?'is G' = {l,a, . . , *|, 
a subgroup of G. As a group is included among its subgroups, 
we have 

Theorem 1. By an adjunction to the domain, the group 
of an equation is reduced to a subgroup. 



300 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV 

Example 1. The group of a;'+x'+x + l = for K{1) is {l, {xiXj)} if 
i= 1 is the rational root. By adjoining i, we obtain the enlarged 
domain R({), for which the group is the identity, since now also Xi=i 
and x=i are in the enlarged domain. The Galoisian resolvent 
|/i_2r+2 for R(\) has the factors Vi-1 in R{i). 

Example 2. The group of x< + l = for i?(l) is G,, given by (12). 
By Ex. 2 of 154, the group for R{i) is G2= fl, (xjXs) (xjX*) } , which is the 
subgroup of Gt to which i=Xi^=Xs^=Xi^=Xi^ belongs. 

These examples illustrate also the important 

Theorem 2. By the adjunction of a rational function 
^{xi, . . . , Xn) of the roots with coefficients in the initial domain 
R, the group of the equation is reduced from G to the subgroup 
H to which 4' belongs. 

It is to be shown that H has the characteristic properties 
A and B ( 149) of the group of the equation for the enlarged 
domain R'. Any rational function <^ of the roots with coef- 
ficients in R' equals a rational function ^i of the roots with 
coefficients in R. 

First, let <f> be unaltered by all the substitutions of H. By 
155, 4>i (which is unaltered by H) is a rational function of 
^ with coefficients in R. Hence </)i = is in R'. Hence property 
A holds for H and R\ 

Second, let equal a quantity p in R', namely, a rational 
function p(\f/) with coefficients in R. Then <^i p(^) is a 
rational function of o^i, . . . , ocn with coefficients in R having 
the value zero, and hence is unaltered by every substitution 
of G and, in particular, by the substitutions of H. The latter 
leave p(^) unaltered and therefore also 0i = <t>. Hence property 
B holds for H and R\ 

EXERCISES 

1. By the adjunction of V2, the group doi x* + l = for R{t) is reduced 
to the identity Gi. 

2. By the adjunction of an imaginary cube root w of unity, the group 
Gt of ac' 2 = for R{1) is reduced to the cyclic group C3. Verify that 
u=Xt/xi belongs to the group C3. By the further adjunction of ^2, 
the group is reduced to the identity d. 

3. Find the group of x*+x^+x^+x+l = for RiVE). 



CHAPTER XV 

SUFFICIENT CONDITION THAT AN ALGEBRAIC EQUATION 
BE SOLVABLE BY RADICALS 

157. Solvability by Radicals. An algebraic equation is 
said to be solvable by radicals if all of its roots can be derived 
by addition, subtraction, multiplication, division, and extraction 
of a pth. root (where p has a finite number of positive integral 
values), these operations being performed a finite number of 
times upon the coefiicients of the equation or upon quanti- 
ties obtained from them by those operations. 

For example, Cardan's formulas (deduced in 158) for the roots of 
x^+CiXC3=0 are A+B, uA+u^B, w^A+uB, where w is an imaginary 
cube root of unity and 

. 3/1 3/1 ^^s c, 

A=\-cz + Vr, B = \-C3Vr, r= 1 , 

\2 \2 27 4 

A being any definite cube root of \c3-\-Vr, and B being chosen so that 
AB= C2/3. The radical in w= j+^V 3 is a square root of the num- 
ber 3 which can be derived from the coefl5cient of 0^ by rational opera- 
tions. 

In the above definition we permitted the use of the opera- 
tion of finding one of the pth. roots of a quantity previously 
determined, but not the use of the operations of finding all of 
the pXh roots. The use of the latter operations would imply 
a knowledge of all of the ;^th roots of unity, whereas we shall 
prove that the pth. roots of unity are expressible in terms of 
V 1 and real radicals. 

The solution of an equation solvable by radicals is often 
accomplished by the use of a series of auxiliary equations, the 
roots of any one of which can be found by rational operations 

301 



302 EQUATIONS SOLVABLE BY RADICALS (Ch. XV 

and root extractions performed upon its coefficients as well as 
upon the coefficients and roots of the preceding equations 
of the series. In order to focus our attention upon a particular 
equation of a series, and to have a general and hence simpler 
phraseology, it is convenient to have the following generaliza- 
tion of the above definition of solvability by radicals. 

An equation with coefficients in a domain R{ki, . . . , km) 
shall be said to be solvable by radicals relatively to R if all of 
its roots can be derived by rational operations and root extrac- 
tions performed upon ^i, . . . , ^ or upon quantities obtained 
from them by those operations. 

For example, *'=2" is evidently solvable relatively to /?(), where 
is a particular imaginary 13th root of unity. 

While a quintic equation whose coefficients Ci, . . . , d are independent 
variables will be shown to be not solvable by radicals, i.e., relatively to 
R{ci, . . . , Ci), it is solvable relatively to R'=R(xi, Ci, . . . , c^), where 
Xi is one root of the quintic (since its group for R' is a solvable group of 
order 24). We have merely shifted the difl&culty to the determination 
of the new domain R'. The benefit that may be gained by the use of R' 
is merely one of phraseology. 

158. Solution of a Cubic Equation. In 

a^cix^+C2XC3=0, 

let Ci, C2, C3 be independent variables. This general cubic 
equation will be discussed from the group standpoint with 
the aim of providing a concrete illustration of the general 
theory which is to follow. 

Let CO be an imaginary cube root of unity. For the domain 
R = R{co, ci, C2, cz), the group of the cubic equation is the 
symmetric group G& on the roots Xi, X2, its ( 153). To the 
cyclic subgroup C3 belongs the function 

b = {xi-X2) {X2 - Xz) {Xz - Xi) . 

By Theorem 2 of 154, 5 is a root of a quadratic equation 
with coefficients in R. In fact, the discriminant of the cubic 
equation is 

52 = cih2^ -\-\%ciC2Cz - 4^2^ - 4ci%3 - 27C32. 



159] RESOLVENT EQUATIONS AND THEIR GROUPS 303 

For the domain R' = {R, d), the group of the cubic equation 
is the cycUc group C3 (156). The substitution (a;iX2a:3) of 
C3 replaces the functions 

with coefl5cients in R, by ocf^\{/ and wx, respectively. Thus 
the substitutions of C3 leave ^ and x^ unaltered. Hence 
( 155) the latter are rational functions, with coefl5cients in 
R, of 5. We have 

^4-x3 = a = 2ci3-9ciC2+27c3, ^-x^= -sV^S, 

^ = |(a-3V^5). 

A cube root rj/ of the last quantity is adjoined to R'; the 
group is thereby reduced to the identity group Gi to which ^ 
belongs. The roots x are now in the enlarged domain {R', yf). 
From the expressions for c\, ^, x, we find by multiplications 

Xi=-\{ci-\-yl^ + x), X2 = \{ci^o^yl^-\-(^x), Xz=\{ci^uyl^-\-(j^x)- 

Here x = (<^i^ 3c2)/^. In brief, the above solution consists 
in finding, by means of a quadratic equation, a function ^ 
which belongs to C3, and then finding, by means of a binomial 
cubic equation, the function ^ which belongs to Gi. 

Taking c\ =0, we obtain Cardan's formulas ( 157). 

159. Resolvent Equations and their Groups. The auxiUary 
quadratic and binomial cubic equations employed in the solu- 
tion * of the general cubic equation are called resolvent equa- 
tions of the latter. In general, let/(ic)=0 be any given equa- 
tion with coefiicients in a given domain R, and let ^ be a rational 
function of its roots with coefficients in R. If ^ belongs to a 
subgroup H of index v under the group G of f{x) = for R, 
we have seen ( 154) that ^ is a root of a resolvent equation 
of degree v with coefficients in R. Suppose that we can solve 
this resolvent equation relatively to R. By adjoining its 
root ^ to the domain R, we obtain a domain Ri = (R, rf) for 
which the group of f{x)=0 is H. If repeated adjunctions 
lead to a domain R,t for which the group is the identity Gi, 

* For brevity, we omit the words " by radicals " after " solution " or "solve." 



304 EQUATIONS SOLVABLE BY RADICALS [Ch. XV 

the roots oif{x)=0 will be in R,, (property A of 149). Hence 
if such a series of resolvent equations can be constructed and 
solved relatively to their domains, the given equation can be 
solved relatively to R. Consequently, we shall discuss the 
question of the solvability of a resolvent equation; to this end 
we must find its group. 

By use of (14) in 154, we proved that ^ is one of v conju- 
gate functions under the group G: 

and that any substitution 5 of G replaces these by 

(2) ^ps, h,s, his, , h^s, 

which are merely the distinct functions (1) rearranged. Hence 
to any substitution 5 of G on the letters xi, . . . , Xn there 
corresponds one definite substitution 

w '=(1 !" !'')-(!" 

on the V letters (1). Similarly, to t corresponds 

since we may rearrange at will the letters in the upper line 
in the two-rowed notation of a substitution. The product 
<7T replaces yj/g^ by ^^^ and hence corresponds to st. 

Theorem 1. The substitutions of G correspond to substitu- 
tions (3) forming a group T. 

The group r is transitive and isomorphic to G. 

Example. Let G be the alternating group on the independent vari- 
ables Xi, . . . , xt. Now \l/={xiX2){x3Xi) belongs formally to the 
group Gt given by (12) of 150. We have 

G=Gi +Gi{xiX3Xi) +Gi(xiXat). 

The indicated substitutions of period 3 replace ^ by 

4'! = (xi Xi) (X4 X2), ypi={xi Xi) (xn xs). 

Since every substitution of Gt leaves also ^2 and ^s luialtered, 

r=}i, (^1^21^3), i^rj^3^2)\. 



1591 RESOLVENT EQUATIONS AND THEIR GROUPS 305 

The importance of the group r is due to 

Theorem 2. r is the group for R of the resolvent equation 

(4) giy)=iy-H)(y-H) (y-^,)=o. 

To prove that r has the characteristic properties A and B 
of the group of (4) for R, note that any rational function 
p(^i, . . . , \f/,) with coefficients in R equals a rational function 
r{xi, . . . , Xu) with coefficients in i?: 

(5) p(^i, . . . , ^f^,)-r{xl, . . . , Xn)=0. 

Since this difference equals a quantity in R, it is unaltered 
by any substitution s oi G on xi, . . . , Xn. Since s gives 
rise to a substitution tr of r on ^i, . . . , ^,, we have 

(6) Pa(\^i, . . . , 4',)-rs{xi, . . . , Xn)=0. 

First, let p be unaltered by every substitution of T, so 
that p = p<r, for every <r in r. Then, by (5) and (6), rs=r for 
every 5 in G. Hence, by property A for the group G, r is in 
the domain R. This proves property A for the group T. 

Next, let p be in R. Then, by (5), r is in R. Hence, by 
property B for the group G, r=rs for every s in G. Then, 
by (5) and (6), p = pa for every o- in r. This proves property 
B for the group r. 

EXERCISES 

1. SiDce r is transitive, (4) is irreducible in R. 

2. If G is the symmetric group on iCi, . . . , Xt, the group r on the 
three y's of Example 2, 150, is the symmetric group of order 6. 

3. If G is the symmetric group on Xi, xi, 0C3, and ^ is the alternating 
function, r is of order 2. 

The function \l/g belongs to the subgroup g'^Sg of G (9). 
Hence the v conjugate functions (1) belong to a complete set 
of conjugate subgroups of G: 

B, g2-^Hg2, g3-^Hg3, . . . , g,~^Hg,. 

In case these groups are all identical, 5" is an invariant 
subgroup of G. In this case, the substitution (3) is the identity 
if s is in H, since 5 then leaves unaltered each ^^^; while any 
substitution s of G and any product hs in which hh'vaE corre- 



306 EQUATIONS SOLVABLE BY RADICALS [Ch. XV 

spond to the same substitution (3) of r. Hence, by (14) in 154, 
we obtain the p distinct substitutions of r by taking those which 
correspond to s = l, g2, . . , g,- We thus have 

Theorem 3. If B is an invariant subgroup of G of index 
V, the group T is a transitive group of order v on v letters and hence 
is regular. 

Corollary. // H is an invariant subgroup of G of prime 
index v, then T is a regular cyclic group of order v. 

This is illustrated by the above example. 

Beginning with the group G of the given equation for the 
given domain R, we can find a series of groups G, H, K, . . . , 
Gi, terminating with the identity group Gi and such that each 
is a maximal invariant subgroup of the preceding. If v is 
the index of H under G, p the index of K under H, etc., the fac- 
tors of composition of G are v, p, . . . 

Construct a rational function \l/ of the roots with coef- 
ficients in R such that 4' belongs to the subgroup H of G. Then 
)^ is a root of an equation of degree v whose group r for R is 
simply isomorphic with the simple quotient group G/H. After 
the adjunction of the root ^ to R, the group of the given equa- 
tion becomes H for the domain {R, xf/) . 

Construct a rational function x of the roots with coef- 
ficients in (R, \f/) such that x belongs to the subgroup K of H. 
Then x is a root of an equation of degree p whose group for 
{R, \f) is simply isomorphic with the simple group H/K. After 
the adjunction of x, the group of the given equation is K. 
Finally, we adjoin a function belonging to Gi and cbtain a 
domain containing xi, . . . , Xn. We therefore have 

Theorem 4. The solution of an equation with the group 
G for the domain R can be reduced to the solution of a series of 
equations each with a simple regular group for the domain obtained 
by adjoining to R a root of each of the earlier equations of the series. 
If, in particular, G is a solvable group, each auxiliary equation 
has a regular cyclic group of prime order. 

160. Equations with a Regular Cyclic Group. To supple- 
ment the last theorem we need the result that any equation with 
a regular cyclic group of prime order p is solvable by radicals. 



1601 EQUATIONS WITH A REGULAR CYCLIC GROUP 307 

We shall now prove this with the restriction that the domain con- 
tains an imaginary pih root e of unity. This restriction will 
be removed in 164 by proving that e can be found by root 
extractions. 

Theorem. An equation, whose group G for a domain R 
containing an imaginary pth root e of unity is a regular cyclic 
group of prime order p, is solvable by radicals relatively to R. 

Let Xii, xi, . . . , Xp-i be the roots of the equation and let 
G be generated by the substitution s = (xoXi . . . Xp-i). Then 
s replaces the function 

with coefficients in R, by e'^di. Let Qi = dt^. Then e< is 
unaltered by 5 and is therefore in R. Thus di is one of the 
pth roots VOi of a quantity in R. Also the sum of the roots 
is given by a coefficient of our equation. Thus 

X<i+Xi-\-X2 + . . . i-Xp-i=c, 

:ro + eJCi+e2x2 + . . . +e''-'^Xp-i = VQ'i, 

Xo+e^Xi-]-^X2 + . . . -{-t^''-''xp-i = VQ^, 

Xo-\-^-^Xi-\-e'^''-'^X2-\- . . . +e^p-i)(p-:rp-i = -^e,-i. 

Multiply these equations by 1, "', t~^^, . . . , -<^-"-', 
respectively, add and apply 

Dividing the resulting equation by p, we get Lagrange's formulas 

Xt=^\c-\-t-'</e[-\-t-^'<^2+ . . . -{-e-^'-'^'^^e^i] 
P 

0'=o, 1, . . . , p-i). 

Since the x's are distinct by hypothesis, the O's are not 
all zero. Thus a certain dt is not zero. Since e may be taken 
as a new e, we may set 0i5^O. While the first radical may 



308 EQUATIONS SOLVABLE BY RADICALS [Ch. XV 

be chosen as any one of the ^th roots of Gi, the remaining radi- 
cals are then fully determined. We have 

where the final factor is in R, being unaltered by s. 

This proof is illustrated by the final work on the cubic 
equation (158). 

161. Cyclotomic Equations. It remains to treat the equa- 
tion 

(7) x''-'^-^x^-^-\- . . . -hx-\-l=0 

for the imaginary pth roots of unity, where p is an odd prime. 
For p = 5, the roots of (7) may be arranged in the order 

For any prime p, it is shown in the theory of numbers that there 
exists a primitive root g oi p such that 

1, g, g', - , g'-\ . 
when divided by p, give in some order the remainders 

1, 2, 3, ... , p-h 
Thus the roots of (7) may be written in the order 

Xi = e, X2 = i', x-i = ^\ . . . , Xp-i = e''^~^. 
Hence 

(8) a:2=:J^l^ x-2=X2'', . , Xp-x=xPp-2, xi=o(fp-i, 

the last relation following from Fermat's theorem that g^~'^ 
is of the form \-\-kp, where k is an integer. 

Consider any substitution s of the group G of (7) for R{\) : 

Xi X2 Xs . ' . Xp-i 



By (8) and property B of 149, it follows that 

But 

by (8). Hence 



1621 CYCLOTOMIC EQUATIONS 300 

provided the symbol Xp be replaced by xi. Thus 

h = a+l, c = h-\-l=a-\-2, . . . (mod/>-l), 

^^M X2 X^ . . . Xj,-x \ 

\Xa Xa+l Xa+2 Xa+p-2/ 

in which Xt+p-i is to be replaced by Xt. Hence s is the power 
a I of (a:i::2:x:3 . . . Xp-i), so that G is a subgroup of the 
cyclic group generated by that substitution. 

After proving in 163 by means of Gauss' lemma that 
equation (7) is irreducible in R{1), we shall know that G is 
transitive (150) and h^nce have the important 

Theorem. // p is an odd prime, the group for the domain 
of rational numbers of the cyclotomic equation for the imaginary 
pXh roots of unity is a regular cyclic group of order p \. 

162. Gauss' Lemma. If a polynomial f{x) with integral 
coefficients, that of the highest power of x being unity, is the product 
of two polynomials with rational coefficients, 

<t>{x)=xr-\-biC(r-''+ . . . +6, ,A(:^)=x+ciX-'+ . . . +c, 

then these coefficients are integers. 

Let the fractions bi, . . . , bm be brought to the least 
positive common denominator /3o and set &< = ft//3o. Then 
/3o, . . . , /3m have no common divisor exceeding unity. Sim- 
ilarly, let Ci = yi/yo, where to, , Tn are integers with no 
common divisor > 1. Multiplying /=0^ by /3o7o, we get 

(9) ^oyof{x) = 4>i(x)-Mx), 

where 

^i=/3o^+/3iX~-*4- . . . +/3, ^i=7oa:"+7iX*-*+ +7. 

We shall assume that j8o7o> 1 and prove that a contradic- 
tion results from this assimiption. Let /> be a prime divisor 
of /3o7o. Since p divides each eoefl&cient of the left member 
of (9), it divides each coefficient of the product <t>i^i. Let 
^i be the first coefficient in <i>i{x) which is not divisible by />; 
let 7t be the first y not divisible by p. The total coefficient 
of x"+''~*~* in <f>-i}l/i is 

. . . -f/3+27t-2+/3+i7t-i-|-/3i7t-|-/3-i7jt+i4-/3i-27+2+ 



310 EQUATIONS SOLVABLE BY RADICALS [Ch. XV 

Since A-i, ^-2, . and 7*-i, 7t-2, ... are divisible by p 
and j8<7t is not, and yet the preceding sum must be divisible 
by p, we have a contradiction. Hence /3o = 7o = l. 

163. Irreducibility of the Cyclotomic Equation. To prove 
that the function f{x), defined by the left member of (7), is 
irreducible in the domain of rational numbers, it is sufficient 
in view of Gauss' lemma to show that f{x) is not the product 
of two polynomials <l){x) and \J/{x) with integral coefficients, 
each having unity as the coefficient of the highest power of x. 
Kronecker's first proof of this fact is essentially as follows: 
Suppose that such a factorization f{x) = </>(if) ^(a:) is possible. 
Taking x = l, we get p = (f>{l) \J/{i) . Since /> is a prime, one 
of these integers, say <^(1), has the value 1. Since the factor 

*0(x) vanishes for at least one of the roots e, e^, . . . , ^~^ 
of f{x) = 0, where e is any one of the roots, we have 

</,() </>(e2) . . . 0(cP-^)=O. 

In other words, the function 

P{x) = <t>{x)-4>{x'^) . . . 0(jc^-^) 

vanishes when x is replaced by any one of the roots of f{x) = 0, 
and hence has the factor /(x). Thus 

P{x)=^f{x)-q{x), 

where q{x) is a polynomial with integral coefficients ( 162, 
or from the fact that the leading coefficient of the divisor f{x) 
is unity). 

Taking x = \, and noting that /(I) =p, we get 

Since this is impossible, /(ic) is irreducible in R{\). 

164. Sufficient Condition for Solvability by Radicals. An 
equation having a solvable group for the domain defined by 
the coefficients is solvable by radicals. We shall prove the more 
general 

Theorem. An algebraic equation having a solvable group 
for any domain R containing the coefficients is solvable by radicals 
relatively to R. 



164] SUFFICIENT CONDITION FOR SOLVABILITY 311 

This will follow if proved for the case of a regular cyclic 
group of prime order. Assuming the theorem for this case, 
let f{x) =0 be an equation whose group for R has the prime 
factors of composition f, p, . . . As in the proof of Theorem 
4 of 159, there is a series of equations w(^) =0, r(x) =0, . . . 
of prime degrees v, p, . . . , the solution of which is equivalent 
to the solution of J{x)=Q. The group for R of (^)=0 is a 
regular cyclic group of prime order v so that this auxiliary 
equation is solvable by radicals relatively to R. The coef- 
ficients of r(x) = are in the domain R' = {R, ^) and its group 
for R' is a regular cycUc group of prime order p; hence it is 
solvable by radicals relatively to R'. In view of the earlier 
result, this second auxihary equation is solvable by radicals 
relatively to R. A repetition of this argument shows that 
f{x)=Q is solvable by radicals relatively to R. 

It remains only to prove that an equation C{x)=0 having 
a regular cyclic group G of prime order p for a domain R is 
solvable by radicals relatively to R. This is true for p = 2. 
To proceed by induction, suppose that every equation having 
a regular cyclic group of prime order <p for any domain D 
is solvable by radicals relatively to D. As in the proof above, 
this implies that the equation for the imaginary ^th roots of 
unity is solvable by radicals (i.e., relatively to the domain of 
rational numbers). In fact, its group for that domain is a 
regular cyclic group of order p \ ( 161), each of whose factors 
of composition is a prime <p. 

Adjoin to R an imaginary p\h root of unity. The group 
of C{x) = for {R, e) is either the initial cyclic group G or the 
identity group. In the latter case, the roots are in {R, e) 
and can be found from the quantities in R by rational opera- 
tions and root extractions, since e was shown to be derivable 
from the rational number by those operations. In the former 
case, C{x) =0 is solvable ( 160) by radicals relatively to {R, e) 
and hence, as before, relatively to R. Hence the induction is 
complete. 

Corollary. // p is an odd prime, the equation for the 
p \ imaginary pth. roots of unity is solvable by radicals. 



312 EQUATIONS SOLVABLE BY RADICALS [Ch. XV 

The theorem implies that any cubic equation is solvable by 
radicals, since its group for any domain containing the coeffi- 
cients is solvable. 

165. Solution of a Quartic Equation. Let the coefficients 
in 

0(!^-\-ax^-\-bx^-\-cx+d = O 

be independent variables, so that we have the general quartic 
equation. Its group for the domain R = R{o3, a, b, c, d), where 
w2 -|- CO -|- 1 =0, is the symmetric group G24 on the roots ici, . . . , 
x^ ( 153). It is a solvable group, having the factors of com- 
position 2, 3, 2, 2. In view of the last theorem, the equation 
is solvable by radicals. We shall give a solution which will 
illustrate the developments of the general theory as presented 
in the next chapter. In fact, we shall employ only binomial 
resolvents. 

To the invariant subgroup G12 composed of the even substi- 
tutions belongs the function 

8 = {XiX2){xi-X3)iXlX4:){X2Xs){X2X4){X3X4), 

whose square equals * the discriminant of the quartic: 

62 = 256(/3-27/2), 

,_ ,_a , _6^ T_bd_ c^ _a^d abc b^ 
4 12' ~~Q I6~T6 l8~216* 

The group of the quartic equation is G12 for {R, d). Employ 
the notations of Example 2, 150. To the invariant subgroup 
G4 of G12 belongs 

0=yi + w3'2 + w2y3, 

whose conjugates under G12 are co<^ and w^^. Thus </> is a root 
of the resolvent cubic 2^ <l>^ = 0. To find <f>^, set 

\l/=yi-\-i>Py2-\-o}y3. 
Then 

^3_^3 = 3('^_^,2)(-yj_y2)(>'2->'3)(3'3-yi)=-3(co-a>2)6, 

^-\-<t>^ = 2{yi^-\-y2^-^y3^)-\-12yiy2y3-\-3(o>-\-or')v, 

* Dickson, Elementary Theory of Equations, p. 42, Ex. 7. 



165] SOLUTION OF A QUARTIC EQUATION 313 

where 

V =yi^y2 -\-yiy2^+yi^y3-hyiy3^ +>'2^3'3 +y2>'3^. 
But 

C^yiY =^v-\-Qyiy2y3-\-^yi^, 

Thus 

^ + 03 = 2(2>'i)'-9(2^O i^yy^) +27yi3;2>'3 
= 263 - Qbiac - M) +27(c2 4-a2(f - ^bd) = - 432/, 
as seen from the ^'-cubic in the example referred to. Hence 

After the adjunction of <l>, the group is G4. Since yi, y2 
and y3 are unaltered by G4, they are in the new domain {R, 
8, 4>). To find them, note that 

<t>^ = ^yi'+(<^+^')^yiy2={^yiy-^^yiy2 = i2l, 

12/ 

yi+y2-\-y3 = b, yi-\-o)y2-]-o}^y3 = <t>, yi-]-oPy2-\-o}y3=-. 

Hence 

The square of h=xi-\-X2X3X4: is 

(^ a;i)^ 4^a;ia;2+4yi=a^ 46+4yi- 

Adjoining a root of 

h^ = a^-^b+4yi, 

we obtain a domain (R, 8, 0, ^1) for which the group of the 
quartic equation is G2 = {1, ixiX2){x3X4)\- Adjoining also a 

root t2 = Xi+X3X2X4 of 

t22 = a^-4b-\-4y2, 



314 EQUATIONS SOLVABLE BY RADICALS [Ch. XV 

we have a domain for which the group is the identity Gi, to 
which therefore xi, . . . , xa belong. To find them, note that 

h^ = a^ 4b+iys, t3=Xi-\-Xi-X2-X3, 

as shown by setting y = b a^/4: in the cubic function in the 
example referred to, which equals n(y yj). To determine 
the sign of the square root, take a;i = l, Xg = 0{g>l); thus 

ht2h= 8c+4a& a^. 

Subject to this condition, the roots are given by 

Axi = -a+h-\-t2+h, ^2=-a-\-h-t2-h, 

4iC3= a h-\-t2 h, ^X4:= a h t2-\-h. 

The above solution can be modified indefinitely, since 
there are infinitely many functions belonging to a given sub- 
group of the equation. Moreover, we might employ other 
subgroups of order 2 of G4 instead of G2. 

EXERCISES 

1. After reducing the group to G2, adjoin j=\/^ and find the quad- 
ratic equation for Vxi X2 + ixi ixi'. 



V^={Xl-0Ciy-{X3-X^y+2i{Xi-Xi){xz-X^) 

-8c o 2 
h 3 



, 4ofe-8c-o' 2 ^/ 12/\' 



For W=Xix-iix3-\-iXi, we have 

VW={x,-Xiy-\-{x3-x,y=TiXi''-2y,=a^-'lh-2yi, 

X,, Xi=\{-a+h{V+W)], X,, X3=-H-o-/. ^-'(F-Pr)]. 

The theoretical interest of this solution is that it leads to a 24- valued 
function V by means of a chain of binominal resolvents. 

2. After reducing the group to G2, we may find Xi and x^ by solving a 
quadratic equation with coefficients in (R, 5, 4>, h) : 

x^+h{a-ii)x+hi-ihayi-c)/ii = 0. 

In terms of Xi, Xt and the quantities known, we may find x, Xti 

ytyi 



Xi-\-Xi = Xi+X2 ti, X3 Xi = ' 



X\ X% 



CHAPTER XVI 

NECESSARY CONDITION THAT AN ALGEBRAIC EQUATION BE 
SOLVABLE BY RADICALS 

166. Galois' Criterion. An algebraic equation is solvable by 
radicals if and only if its group for the domain defined by the 
coeffixients is a solvable group. 

It is occasionally useful to employ the generalization: 

An equation is solvable by radicals relatively to a domain R 
containing the coefficients if and only if its group for R is solvable. 

That the solvability of the group is a sufficient condition 
for the solvability of the equation was proved in 164. We 
shall now prove that it is a necessary condition. By hypoth- 
esis the roots xi, . . . , ocn oi the equation can be derived 
by rational operations and root extractions from quantities 
in the domain R = R{ki, . . . , km) or from quantities obtained 
from them by those operations. The index of each root ex- 
traction may be assumed to be prime, since z"' = a is equivalent 
to the pair of equations z^=w, lif =a. li ^, r], . . . , \j/ denote 
the radicals which enter the expressions for a:i, . . . , Xn, the 
solution may be exhibited by a series of binomial equations 

|^=L(^l, . . . , km), r = Mi^, kl, . . . , km), . . . , 

V=S{. . . , V, ^, kl, . . . , k^) 

of prime degrees \ n, . . . , &, together with equations which 
express xi, . . . , Xn rationally in terms of ^, . . . , ^, ^i , . . , 
Here L, . . . , S are rational functions with rational 



coefficients of their arguments. 

Consider, therefore, a binomial equation of 

(2) x''-A=0, 

315 



prime degree 



316 EQUATIONS SOLVABLE BY RADICALS [Ch. XVI 

where A is in the domain R. Let be an imaginary p\h root 
of unity. If one root r of (2) belongs to the domain R' = {R, c), 
all of the remaining roots er, e^r, . . . , ^"V belong to R' 
and the group of (2) for R' is the identity. In the contrary 
case, A is not the />th power of a quantity in R', and (2) is irre- 
ducible m. R' ( 143). The notation of the roots can be chosen 
so that 

.By an argument like that of 161 with p 1 replaced by p, 
we see that the group of (2) for R' is a subgroup of the cyclic 
group generated by (jcia!;2^3 . . . Xp). Since (2) is now irre- 
ducible in R', its group is transitive and hence of order = p. 

For a domain containing A and an imaginary pth root of unity, 
the group of the binomial equation (2) of prime degree p is the 
identity group if any root is in the domain, but is a regular cyclic 
group of order p if no root is in the domain. For examples, see 
150. 

Any one of the binomial equations (1) of prime degree p is 
equivalent to a series of equations each having a regular cyclic 
group of prime order for a certain domain. We include in the 
series the equations having regular cyclic groups of prime 
orders dividing p l, which together serve to determine an 
imaginary pth root e of unity ( 161, Theorem 4 of 159). 
After the adjunction of e, the group of the binomial equation 
was just proved to be either the identity group or a regular 
cychc group of order p. In the former case, the desired series 
of equations is the one previously defined; in the latter case, 
that series together with the given binomial equation. Thus 
the set of binomial equations (1) is equivalent to a series of 
equations of prime degrees, each having a regular cyclic group 
for the domain symbolized in the same line : 

<^(y; ki, . . . , k)=0, R = R{ki, . . . , hi)] 

rp{z; y, ku . . . )=0, (y, R); 

e{w; . . . z, y, ki, . . . )=0, (. . . 2, y, R). 



1671 THEOREMS OF GALOIS, JORDAN AND HOLDER 317 

Each of these equations is solvable by radicals relatively to 
the corresponding domain ( 164). Adjoin one root y of <> = 
to R\ the group is now a subgroup R of the initial group G, 
including the possibiUty that E.=G (156). Solve ^ = and 
adjoin one root z to the domain {y^ R); the group is now a 
subgroup of H. Proceeding in this manner, we finally reach 
the domain {w, . . . , z, y, R) containing each root Xt of the 
proposed equation, whose group is therefore now the identity 
group Gi. 

The theorem of Galois states that, by each of these adjunc- 
tions, the group of the proposed equation is either not reduced 
at all or else is reduced to an invariant subgroup of prime 
index. This theorem will be derived as a corollary in the next 
section from a theorem of which other important apphcations 
will be made later on. Hence the distinct groups G, H, . . . , 
Gi, obtained by the successive adjunctions, form a series of 
composition of G with only prime factors of composition. Thus 
G is a solvable group. We therefore have Galois' criterion 
for the solvability of an equation. 

For w>4, the factors of composition of the symmetric 
group on n letters (17) are 2 and ^nl, the latter of which is 
not prime. By 153 the group of the general equation of 
degree n, i.e., one whose coefficients ci, . . . , Cn are independent 
complex variables, is the symmetric group when the domain 
is that defined by ci, . . . , c and a finite number of constants. 
We therefore have the 

Theorem. The general equation of degree n>^is not solvable 
by radicals * Moreover, its roots cannot be found by rational 
operations and root extractions perfortned upon the coefficients 
and any constants, finite in number, or upon quantities obtained 
from them by those operations. 

167. Theorems of Galois, Jordan and Holder. Of prime 
importance is 

Jordan's Theorem.! Let the group Gi for a domain R of 

* Ruffini, Teoria generde ddle equazioni. . . , Bologna, 1799. N. H. Abd, 
(Euvres, vol. 1, 1881, pp. 66-94. 

t Jordan, TraiU des substitutions, p. 268-9. 



318 EQUATIONS SOLVABLE BY RADICALS (Ch. XVI 

an algebraic equation Fi{x)=0 be reduced to G'l by the adjunc- 
tion of all of the roots of a second equation Foix) =0, and let the 
group G2 for R of the second equation be reduced to G'2 by the 
adjunction of all of the roots of tlie first equation. Tlien G'\ and 
G'2 are invariant subgroups of G\ and G2, respectively, of equal 
indices, and * the quotient-groups Gi/G'i and G2/C2 are simply 
isomorphic. 

By 154 there exists a rational function ^1 with coefficients 
in R of the roots ^1, . . , ^i, of the first equation, such that 
^1 belongs to the subgroup G\ of Gi. Since the adjunction 
of the roots rji, . . . , ?; of the second equation reduces Gi to 
G'l, property A of G'l ( 149) shows that ^1 Ues in the enlarged 
domain : 

(3) ^l(h, . , $) = 01(171, . . . , Vm), 

where <t>i is a rational function with coefficients in R. 

Let ^i, \l/2, . . , ^t denote all of the numerically distinct 
values which \f/i can take under the substitutions (on ^1, . . . , 
^n) of Gi. Then G'l is of index k under Gi ( 154, Theorem 2). 
Let 01, . . . , 0j denote all of the numerically distinct values 
which 01 can take under the substitutions (on 771, ... , 7?^) 
of G2. The k quantities rp are the roots of an equation irre- 
ducible in R; likewise for the / quantities 0. Since these two 
irreducible equations have a common root i/'i = 0i, they are 
identical ( 144). Hence the ^'s coincide in some order with 
the 0's; in particular k=l. 

If Si is a substitution of Gi which replaces ^1 by ^1, then 
Si transforms the group G'l of \}/i into the group of ^i of the 
same order as G'l. Since xf/i equals a 0, it is in the domain 
R' = {R, 771, . . . , Tjm) and hence is unaltered by the substi- 
tutions of the group G'l of Fi{x) = for that domain R' ( 149, 
property B). Hence the group of ^< contains all of the sub- 
stitutions of G'l and, being of the same order, is identical with 
G'l. Thus G'l is invariant in G\. The group for R of the 
irreducible equation satisfied by ypi is therefore the quotient 
group Gi/G'i ( 159). 

* This supplement and the proof here employed arc due to HOlder, Mathe- 
matische Annalen, vol. 34, (1889), p. 47. 



1671 THEOREMS OF GALOIS, JORDAN AND HOLDER 319 

Let H2 be the subgroup of G2 to which 01(171, . . . , ij,) 
belongs. Since 4>i is a root of an equation of degree l = k irre- 
ducible in R, the group H2 is of index k under G2. By the 
adjunction of </>i, i.e., of ^1 by (3), the group G2 of F2{x)=Q 
for R is reduced to ^2 ( 156, Theorem 2). If not merely 
^1(^1, . , l), but all of the $'s themselves be adjoined, the 
group G2 reduces perhaps further to a subgroup of H2. Hence 
G'2 is contained in E2. We thus have the preliminary result: 
If the group of Fi(a:)=0 reduces to a subgroup of index k on 
adjoining all of the roots of F2{x)=Q, then the group of F2{x) =0 
reduces to a subgroup of index ^1, ^1 ^ ^, on adjoining all of the 
roots of Fi(x)=0. 

Interchanging Fi and F2 in the preceding statement, we 
obtain the result: If the group of F2{x) =0 reduces to a subgroup 
of index ki on adjoining all the roots of Fi{x) =0, then the group 
of Fi{x)=Q reduces to a subgroup of index ^2, ^2^^i, on ad- 
joining all the roots of F2{x) =0. Since the hypothesis for the 
second statement is identical with the conclusion for the first 
statement, it follows that 

k2 = k, ki = k, ^2 = ^1, 

so that ki = k. Hence the group G'2 of the theorem is identical 
with the group H2 of all of the substitutions in G2 which leave 
<f>i unaltered. For the same reason that G'l is invariant in Gi, 
it now follows that G'2 is invariant in G2. The equation irre- 
ducible in R and satisfied by 0i has as its group the quotient- 
group G2/G'2. 

Since the two irreducible equations in R satisfied by <f>i 
and \f/i, respectively, were shown to be identical, their groups 
Gi/G'i and G2/G'2 differ only in the notations employed for the 
letters on which they operate, and hence are simply isomorphic. 

We shall derive as a corollary 

Galois' Theorem. By the adjunction of any one root of an 
equation 7^2 (x) =0 whose group for R is a regular cyclic group of 
prime order p, the group for R of the equation Fi (x) = either 
is not reduced at all or else is reduced to an invariant subgroup oj 
index p. 



320 EQUATIONS SOLVABLE BY RADICALS [Ch.XVI 

In fact, by adjoining one root xi of F2(x)=0, we adjoin 
all of its roots, since each is a rational function of xi with coef- 
ficients in R ( 155). For, the identity is the only substitution 
of the group for R of F2{x)=0 which leaves xi numerically 
unaltered. 

We shall state certain results not presupposed in what follows. A 
brief argument (cf. Dickson's Theory of Algebraic Equations, 1903, p. 83) 
now leads to Abel's theorem: The roots of an equation solvable by radicals 
can be given such a form that each of the radicals occurring in the expres- 
sions for the roots are expressible rationally in terms of the roots of the 
equation and certain roots of unity. This was proved by Abel by a long 
algebraic discussion without the aid of groups and employed in his proof 
of the impossibility * of solving by radicals the general equation of degree 

For a domain R an irreducible equation of prime degree whose roots 
are all rational functions of two of the roots with coefficients in R is called 
a Galoisian equation. Galois proved that it is solvable by radicals and that 
every irreducible equation of prime degree which is solvable by radicals 
is a Galoisian equation. For a detailed exposition with illustrative exam- 
ples, see Dickson's Theory of Algebraic Equations, 1903, pp. 87-93. 

A cubic equation having three real roots cannot f be solved by real 
radicals (the " irreducible case "). 

* Cited in 166. Cf. Scrret, Cours d'Algibre supirieure, ed. 4, vol. 2, pp. 
497-517. 

t H. Weber, Algebra, ed. 2, 1898, vol. 1, 657; Kleines Lehrbuch der Algebra, 
1912, p. 381. 



CHAPTER XVII 
CONSTRUCTIONS WITH RULER AND COMPASSES 

168. Some Celebrated Problems of Greek Origin. In the 

Dalian problem of the duplication of a cube, we are given the 
length s of an edge of a cube and seek to construct by ruler 
and compasses the edge x oi a. cube whose volume is double 
that of the first cube. For this problem, as well as for the 
problem of the trisection of an arbitrary angle, and for the 
problem of the construction of a regular polygon of 7 or 9 
sides, the ancients sought in vain for constructions by ruler 
and compasses. The impossibility of these constructions 
was proved only in recent times. To the analytic methods 
employed in the proof of this impossibility is due also the 
discovery of new constructions, such as that for the regular 
polygon of 17 sides, the constructibiUty of which was not sus- 
pected during the twenty centuries from Euclid to Gauss. 

169. Analytic Criterion for ConstructibiUty by Ruler and 
Compasses. The first step in our treatment of the problems 
mentioned in 168 is their analytic formulation. In the 
Delian problem, we are led at once to the equation j^ = 2s^. 
Next, if angle 120 could be trisected or if a regular polygon 
of 9 sides could be constructed by ruler and compasses, angle 
40 could be constructed and hence cos 40. In the identity 

cos SA=4: cos^ AZ cos A , 
take ^ = 40 . Since cos 1 20 = - , we get 

4 cos3 40 - 3 cos 40 -|-^ = 0. 

Multiply by 2 and set a: = 2 cos 40; thus 

(1) r-3a;+l=0. 

321 



322 CONSTRUCTIONS BY RULER AND COMPASSES [Ch. XVII 

In this problem and- the Delian problem, we are given the 
coefficients of a cubic equation and ask whether or not a line 
whose length is a root x can be constructed by ruler and com- 
passes. We shall first prove that an affirmative or negative 
answer is to be given according as x can or cannot be derived 
from the coefficients by rational operations and extractions 
of real square roots. 

For any proposed construction we are concerned with 
certain numbers, some expressing lengths, areas, etc., others 
being the coordinates of points, and still others being the coef- 
ficients of equations of straight lines or circles referred to rect- 
angular axes. We shall establish the 

Criterion. A proposed constrtiction by ruler and compasses 
is possible if and only if the numbers which define analytically 
the desired geometrical elements can be derived from those defining 
the given elements by rational operations and extractions of real 
square roots performed a finite number of times. 

First, let the construction be possible. The straight lines 
and circles drawn in making the construction can be located 
by means of points either initially given or obtained as the 
intersections of straight lines and circles. The coordinates of 
the intersection of two intersecting lines are evidently rational 
functions of the coefficients of the equations of the lines. If 
the straight line y = mx-\-b intersects the circle 

the coordinates of the points of intersection are found by elim- 
inating y, solving the resulting quadratic for x, and inserting 
the roots x into y = mx-\-b. Hence the coordinates are found 
from m, b, p, q, r by rational operations and the extraction of 
a single real square root. Finally, two intersecting circles cross 
at the intersections of one of them with their conunon chord, 
so that this case reduces to the preceding. 

That the criterion gives also a sufficient condition for con- 
structibility is shown by the facts that the sum or difference 
of two segments of straight lines can be found by use of com- 
passes, that p = ab can be found by constructing p in 1 i a = b : p 



$ 1721 REGULAR POLYGONS 323 

by use of parallels, and similarly q = a/h in 1 : b = q : a. Finally, 
if n is a positive number, V can be constructed by the use of 
a semicircle of diameter l+ and a perpendicular at the point 
separating the segments of lengths 1, n. 

170. Trisection of an Angle. To prove that it is impossible 
to trisect an arbitrary angle by ruler and compasses, it suffices 
to prove that angle 120 cannot be trisected. We saw that 
2 cos 40 is a root of (1). In the domain of rational numbers, 
Eq. (1) is irreducible ( 144, Ex. 3) and has the discriminant 
81 ; hence its group is of order 3. By the adjunction of a square 
root, the group is either not reduced at all or is reduced to a 
invariant subgroup of index 2 ( 167, Galois' Theorem). Hence 
no such reduction is possible in the present case. If therefore 
the cubic had a constructible root, its adjunction would cause 
no reduction of the group, whereas the adjunction of any root 
reduces the group to the identity. 

171. Duplication of a Cube. If an edge of the cube be taken 
as the unit of length, the edge of the desired cube is a root of 

For the domain of rational numbers this irreducible equation 
has as its group the symmetric group Ge. The adjunction of 
any root reduces it to a group of index 3. Hence no root can 
be found by extractions of square roots. 

172. Regular Polygons. The construction of a regular 
polygon of n sides by ruler and compasses is equivalent to that 
of angle 2ir/n and hence of a line of length cos 2ir/w. The 
irreducible equation with rational coefficients satisfied by the 
latter number is much more difficult to form and treat than that 
with the root 

(2) f = cos h* sin , 

n n 

where i = Vl. In view De Moivre's theorem, r is an nth 
root of unity. Moreover, 

1 2X . . 27r ,1 o^^e^T 

- = cos tsin . r-\ = 2 cos . 

r n n r n 



324 CONSTRUCTIONS BY RULER AND COMPASSES [Ch. Xvll 

Hence if r can be expressed in terms of i and real square roots, 
cos 2'K/n can be expressed in terms of real square roots. The 
converse is seen to be true by an inspection of (2), since the 
sine can be found from the cosine by a real square root. Hence 
a regular -gon can be constructed by ruler and compasses if 
and only if the th root (2) of unity can be found by the ex- 
traction of square roots, all except the last one of which is real. 
If n is an odd prime ^, r is a root of an equation of degree 
p \ irreducible in the domain R of all rational numbers and 
having as its group for R a regular cyclic group C of order 
/> 1 ( 161-3). The adjunction of any root reduces C to 
the identity. If a regular p-gon can be constructed, the ad- 
junction of the root r is equivalent to that of several square 
roots, the adjunction of each of which causes either no reduction 
in the group or a reduction to a subgroup of index 2. Hence 
a regular ^-gon can be constructed by ruler and compasses 
if and only if p \ is a power 2* of 2. But if /j =Jq, where / 
is odd, then 2''-fl has the factor 2*' + l- Hence a prime of the 
form 2''4-l is of the form 

(3) 22' -fl. 

For t = 0, 1, 2, 3, 4, the corresponding numbers are 3, 5, 17, 
257, 65537, and are all primes. But for / = 5, 6, 7, 8, 9, 11, 12, 
etc., the number is known to be not prime. 

Next, let n = ab, where a and h are relatively prime integers 
> 1. If a regular a-gon and a regular 6-gon can be constructed 
by ruler and compasses, the same is true of a regular w-gon. 
For, multiples of the angles 27r/a and 2Tr/h can then be con- 
structed and hence also the sum of these multiples. Since 
there exist integers c and d such that ca-\-dh = \, the angle 

, 27r 27r 27r. ,, s 27r 

d \-c- =-{dh+ca) = , 

a ab ab 

and therefore also the a6-gon, can be constructed. Conversely, 
from the latter we obtain a regular a-gon by using the 1st, 
(6-fl)th, (26 + l)th, . . . , [(a-l)& + l]th vertices. Hence if 
n = f(l . . . , where p, q, . . . are distinct primes, a regular 



172] REGULAR POLYGONS 326 

n-gon can be constructed if and only if a regular p^-gon, (f-gon, 
. . , can be constructed. A 2*-gon can be constructed by 
repeated bisections of 180. 

It therefore remains only to discuss the regular p^-gon, 
where p is an odd prime. By De Moivre's theorem, 

p = cos .+z sm -, 

P' f 

is a root of x" =1, but not oi x^ =1, and hence is a root of 

(4) ^!!l^:^(p-i)+a:'(i'-2)+ . . . +^_^1=0 (/ = />*-*). 

Since p^, p^", . . . , p'" give the t roots of a:' = l, the remaining 
tp t powers of p, with positive exponents less than tp and not 
divisible by p, are roots of (4) and give all of the roots of (4). 
They are called the primitive />*th roots of unity. 

For p*=9, the six primitive ninth roots of unity are p, p*, p*, p*, p% p' 
and are the roots of x^+x^+l = 0. 

The proof that (4) is irreducible in the domain R of all 
rational numbers differs from that in 163 for the special 
case 5 = 1 only in the detail of having, instead of c, c^, . . . , 
e**"^ in the former case, the roots p, p", p*, . , . , p' of (4), 
where I, a, b, . . . , / denote the positive integers less than />* 
and not divisible by p, and p is an arbitrary primitive p^th 
root of unity. 

As shown in the theory of numbers, there exists a primitive 
root g of f, where p is an odd prime, i.e., an integer g such that 

1, g, g^ . . . , ^-' ik=p'-p'-'), 

when divided by p", give as remainders in some order the posi- 
tive integers less than ^ and not divisible by p. Thus the 
roots of (4) are 

p, p*, p'*, . . . , pT . 

In the former example p' = 9, we may take g = 2. Then the preceding 
roots are p, p*, p*, p', p', p', respectively. 

Since each root of (4) can therefore be expressed as the 
^th power of the preceding root, we readily find as in 161 



326 CONSTRUCTIONS BY RULER AND COMPASSES [Ch. XVII 

that the group of (4) for 2? is a regular cyclic group of order k. 
li s>l, k is not a power of 2, and by the usual argument the 
regular />'-gon cannot be constructed by ruler and compasses. 

Combining our results, we have the 

Theorem. A regular polygon of n sides can he constructed 
by ruler and compasses if and only if n = '^p\p2 , where 
pi, P2, . . . are distinct primes of the form (3). 

Since therefore a regular 9-gon cannot be constructed, we 
have a new proof that angle 120** cannot be trisected by ruler 
and compasses. 

Gauss * was the first to prove that a regular p-gon can 
be constructed if /> is a prime of the form (3) ; he stated,t but 
apparently did not publish a proof of, the remaining part of 
the above theorem. For the elegant method invented by 
Gauss for finding the series of quadratic equations leading 
to a 17th root of unity and the actual geometrical construction 
of a regular 17-gon, as well as for a longer proof of the above 
theorem without the aid of group theory, the reader may con- 
sult the monograph by Dickson, { where references to other 
books are given. 

* Disqidsitiones ArithmetioB, 1801, Art. 335-366 [=Werke, 1]; German 
translation by Maser, 1889, pp. 397-448, 630-6.52. 

t Gauss-Maser, p. 447. 

} Monographs on Modern Mathematics, edited by J. W. A. Young, New York, 
1911. A brief, but more elementary, treatment is given in Dickson's Elementary 
Theory of Equations, 1914, pp. 84-92. A still more elementary discussion is that 
by Dickson, Amer. Math. Monthly, vol. 21 (1914), 259-262. 



CHAPTER XVIII 

THE INFLEXION POINTS OF A PLANE CUBIC CURVE 

173. Homogeneous Coordinates of Points in a Plane. Let 

aiX-\-biy-\-Ci=0 (* = 1, 2, 3) 

be any three linear equations such that 

ai bi c\ 

A= a2 62 C2 ?^0. 

a3 63 C3 

Interpret x and y as the Cartesian coordinates of a point referred 
to rectangular axes. Then the three equations represent three 
straight lines Lt forming a triangle. Choose the sign before 
the radical so that 

_ (hx-\-hiy-\-Ci 

is positive for a point {x, y) inside the triangle, and hence is 
the length of the perpendicular from that point to L^. The 
homogeneous coordinates of a point {x, y) are three numbers 
x\, X2, X3 such that 

pXi=kiP\, pX2 = k2p2, pxs hpsj 

where ki, ^2, ks are constants, the same for all points, while 
p is an arbitrary factor of proportionality. Thus only the 
ratios of xi, X2, X3 are defined. The coefficients of the linear 
function ktpi, which are proportional to Ot, 6<, c<, will hence- 
forth be denoted by those same letters. Then 

(1) pXi = aiX-hbty-\-Cu Aj^^O (t = l, 2, 3). 

Solving these equations by determinants, we get 

Ax = p^AiXi, Ay = p^BtXi A = p^C,a;<, 
327 



328 INFLEXION POINTS OF A CUBIC CURVE [Ch. XVIIl 

where Ai is the cofactor of a in A, 3% that of bt, and d that of 
Ci. Hence 

(2) x^^ ^2^ 

2 Q^ 2 CiOCi 

Thus any equation /(a;, y)=0 can be expressed as a homoge- 
neous equation <i>{x\, X2, a;3)=0 of the same total degree, and 
conversely. In particular, any straight line is represented by 
an equation of the first degree in xi, X2, Xs, and conversely. 
For example, a:i = represents a side of the triangle of reference. 
Let yi, y2, y3 be the homogeneous coordinates of the same 
point (x, y) referred to a new triangle of reference having the 
sides L'i. As before, 

(lO pyi = a\x+b\y+c\ (f = l, 2, 3), 

where the right member equated to zero represents L'i. Solv- 
ing equations (1') as we did (1), we obtain x and y as Unear frac- 
tional functions of yi, y2, ys- Inserting these values into (1), 
we get formulas like 

(3) Xi-=Ciiyi-{-Ci2y2+Ci3y.i, \cij\^0 (z = l, 2, 3). 

Thus a change of triangle of reference gives rise to a linear 
transformation of the homogeneous coordinates. 

Let/(xi, X2, xs) be a homogeneous rational integral function 
of the wth degree. Under the transformation (3), let it become 
<l>(yi) 3'2, y^)- Then </> = represents the same curve as/=0, 
but referred to the new triangle of reference. Let 

t = kxi<'X2^X3'' {a-\-b-\-c = n) 

be any term of/. Then 

Xi = at, X2 = bt, X3 =ct. 

dxi dX2 dX3 

Their sum is nt. Hence we have Euler's theorem: 

(4) x,^-\-x^-i-x;-^ = nf. 

dxi dX2 dxs 



174] 



HESSIAN CURVE 



329 



If xi, X2, X3 is a set of solutions, not all zero, of 

(5) ^ = 0, -^ = 0, -^ = 0, 

^ ^ dXl dX2 3X3 ' 

and hence by (4) of /=0, the point {xi, xi, x-^ is called a singular 
point of the curve /=0. At this point. 



dyj ^ dxi dyj 



C/ = l,2,3). 



Hence the definition of a singular point is independent of the 
special triangle of reference chosen. It is readily proved, 
but not presupposed in what follows, that two or more branches 
of the curve pass through any singular point, which is there- 
fore called a double or multiple point. 

174. Hessian Curve. The Hessian of / is 



h = 



dxi^ 

8x28x1 
ay 



d'f 



dj 



dxi 8X2 

8X2^ 



8X1 8X3 
8X28X3 



8x38x1 8x38x2 8x3^ 

Let transformation (3), of determinant A, replace / by <t>(yi, 
y2, yz)' The product hA is a determinant of the third order, in 
which the element in the ^*th row and yth column is the sum 
of the products of the elements of the ith row of // by the corre- 
sponding elements of the^'th column of A, and hence is 



8Y 



8x<8xi 



cir 



9'/ 



8x,8x2 



C2r 



m 



8x<8x3 



Czj. 



The latter is the partial derivative with respect to x< of 

9/ 8xi 8/ 8X2 8/ 8X3 ^80 
8xi dyt 8x2 dyj 8x3 8y; 8y/ 

Let A' be the determinant obtained from A by interchanging 
its rows and columns. By the same rule of multiplication, 



= Hessian H of 0. 

r. J-l, 2.3 



330 INFLEXION POINTS OF A CUBIC CURVE (Ch. XVIII 

the element in the rth- row and jth. column of the determinant 
equal to A' AA is 

^'dxAdyJ ^'dx2\dyj/ "'dx-sKdyJ dyXdyJ' 
since c, is the partial derivative of Xi with respect to yr. Hence 

I dyrdyj 

In words, A^h becomes // under the transformation (3), 
so that H = represents the same curve as h = 0, but referred 
to the new triangle of reference. Hence there is associated 
with any curve /=0 a definite Hessian curve h = independent 
of the choice of the triangle of reference. 

175. Points of Inflexion of a Cubic Curve. Let/(a;i, X2, xs) 
be of the third degree. Choose a triangle of reference having 
the vertex P = (0, 0, 1) at a point on the curve /=0, not a singu- 
lar point. Then there is no term involving xs^, and the coef- 
ficients of the terms rxiXs^ and SX2X3^ are not both zero, since 
otherwise the derivatives (5) would all vanish at P. Hence 
we may take rxi-^sx2 as a side of a new triangle of reference 
with the same vertex P and obtain 

X3^xi-\-X3{axi^-\-bxiX2-\-cx2^)-\-<t>(xi, 3:2) =0 

as the new equation of our curve. Replacing xs by 

X3-{axi-j-bx2)/2, 
we get 

Fi=xs^xi-{-ex3X2^-^C{xi, X2). 

Denote the second derivative of the cubic function C with 
respect to oci and Xj by Cy. Then the Hessian of Fi is 

Cii C12 2x3 



Hi = 



C21 C22-\-2ex3 2ex2 



= -8e:r33-F 



2x3 2eji;2 2^i 

Hence P = (0, 0, 1) is on Fi =0 if and only if e = 0. 

If d is the coefficient of X2^ in C, then xi ~0 meets Pi = in 
the points for which X2^{ex3-^dx2)=0, and these three points 



175] 



INFLEXION POINTS AND TRIANGLES 



331 



coincide (at P) if and only if e = 0. In that case P is called a 
point of inflexion of Fi = and Xi = {) the inflexion tangent 
toFi=Oati'. 

Thus P is a point of inflexion of Fi =0 if and only if it is on 
ni=0. Hence, by 174, each intersection of a cubic curve 
/=0 without a singular point with its Hessian curve h = is a 
point of inflexion off=0, and conversely. 

There is certainly at least one intersection. For, by elim- 
inating X3 between /=0 and h = 0, we get a homogeneous 
equation in xi and X2, having therefore at least one set of solu- 
tions x'l, x'2. Then, for X\=x'i, iC2=ic'2, the equations /=0, 
/f = have at least one common root x=x'z. Thus {x'\, x'2, 
x'i) is an intersection and therefore a point of inflexion of /=0. 
Taking this point as a vertex (0, 0, 1) of a triangle of reference 
and proceeding as before, we get F of type Fi with e = 0. If 
the coefficient d of X2^ in F is zero, F has the factor x\. But, 
if F=xiQ, the derivatives 

dxi dxi dX2 dX2 dxs dX3 

all vanish at a point of intersection of 0:1 = 0, Q=0, whereas 
F = has no singular point. Hence d9^0. Taking d^X2 as a 
new X2, and then adding a suitable multiple of Xi to X2 to delete 
the term with X2^^i, we get 



F=X3^Xi+C,' C=X2^-^dbx2Xi^-{-axi^, 



H = 2xi 



Cii C12 
C21 C22 



-^X3^C22 = 72Xi 



bx2-{-axi bx\ 

bX\ X2 



-24X32X2. 



Eliminating x^^ between F = 0, ^ = 0, we gt 
xi C 

X2 ^x\{bx2^-\-axiX2 l^x\^) 
= X2*+Qbx2^Xi^-\-^ax2Xi^ - 362x1* = 0. 

If xi=0, then X2 = and the intersection is (0, 0, 1). For the 
remaining intersections, we miay set xi = 1 ; then each root of 

(6) f*-|-6&r2+4ar-362 = 



332 INFLEXION POINTS OF A CUBIC CURVE (Ch. XVIII 

leads to the inflexion points (1, r, j), where 

-52 = C = r3+36r+a. 

If 5 = 0, (6) would have a double root and a^-\-4i^ = 0. But 
the partial derivatives of F all vanish at (1, X2, 0) if r:2^+6 = 0, 
2bx2+a = 0, and hence if b = 0, a[;2 = 0, or if 67^0, X2= a/{2h), 
whereas F has no singular point. Hence 

(7) a2+4JV0 

and (6) has four distinct roots r for each of which S9^0. Thus 
there are exactly nine distinct points of inflexion. 

The two points (1, r, s) with a fixed r are collinear with 
P = (0, 0, 1), being on X2 = rxi. For the remaining roots p of 
(6), we have 

p8 +rp2 + (r2 +66) p+r3 4.6 J;. +4^ = 0. 
The product of this by r can be written in- the form 

r(p3+36p+a) + (rp+^)2 = 0, k = h{r'-\-Sb). 
Hence the quadratic factor in 

(8) ^H-\-rF = (rxi-X2){xs^-krx2+kxiY\ 

vanishes at (1, p, 0-), where 

-<r2 = p3+36p+a. 

Thus the nine points of inflexion lie by threes upon the three 
straight lines given by (8), which are said to form an inflexion 
triangle. There are four inflexion triangles, one for each root 
r of (6). 

The roots of (6) are the only values of r for which H-\-2irF 
has a linear factor /. In fact, / = meets F = in three points 
on n = which are therefore points of inflexion. Thus / has two 
companion linear functions such that llil2 = is one of the 
four inflexion triangles. Hence llil2 = H-)r2ApF, where p is 
one of the roots of (6). By hypothesis, lQ = n-{-2irF. If 



176] GROUP FOR THE INFLEXION POINTS 333 

rj^p, we see by subtraction that F has the factors / and Qlih 
and hence has a singular point, contrary to assumption. 

Corresponding results follow at once for the general cubic 
curve /=0. We saw that/ can be reduced to F by a linear 
transformation of a certain determinant 5. But F is replaced 
by a Uke form by the transformation which multiplies xi, 
X2, X3 by 8~^, 1, 8, respectively, and thus has the determinant 
8~^. The product of the two transformations is of determi- 
nant unity and replaces/ by a form F. Hence ( 1 74) , it replaces 
the Hessian h oi f by the Hessian H of F. Thus for each root 
r of (6) , in which a and b are certain functions of the coefficients 
of/, A+24r/=0 represents an inflexion triangle of/. 

Furthermore, a and b are rational functions of the coefficients 
of/. For, there are exactly four values of r for which (f>=h-\-2Arf 
has a linear factor xi mxonxs. Replacing xi by mx2-\-nx3 
in (f), we obtain a cubic function of X2 and x-s whose coefficients 
must vanish. Eliminating m and n, we obtain two equations 
in which r and the coefficients of / enter rationally and inte- 
grally. The greatest common divisor of their left members 
must be a function of r whose coefficients are rational in those 
of/. The latter is therefore true ( 145, first foot-note) of the 
quartic equation * for r with no multiple root. 

176. Group G of the Equation X for the Abscissas of the 
Points of Inflexion. Let R be the domain defined by the co- 
efficients of the equation /=0 of a cubic curve without singular 
points. We employ a new triangle of reference whose side 
ii;i = does not contain a point of inflexion. This can be ac- 
complished by a Unear transformation on Xi, X2, xs with coef- 
ficients in R. We pass to Cartesian coordinates by setting 
X2lx\=x, xzlx\=y. After applying a transformation with 
coefficients in i?, corresponding to a" rotation of the axes, we may 
assume that the >'-axis is not parallel to any line joining two 
inflexion points of /=0. Then the abscissas a;i, . . . , x of 
the points of inflexion are distinct. By eUminating >^ and y* 
between the equations of the curve and its Hessian curve, we 

* We do not employ the fact, which now follows readily, that the coefficients 
of (6) are rational integral invariants of /. 



Xi 


y* 1 


Xj 


yj 1 


Xt 


yt 1 



334 INFLEXION POINTS OF A CUBIC CURVE [Ch. XVIII 

obtain y expressed as a rational function <j){x) of x with coef- 
ficients in R. In fact, the ordinate of an inflexion point is 
uniquely determined by its abscissa. By substituting </)(x) 
for y in/=0, we obtain the equation X for the nine abscissas 
of the points of inflexion. 

For three collinear points of inflexion, 



= 0. 



Replacing yt by <t>{xi), etc., we obtain a rational relation 

(9) \l^{xi, xj, xi)=0, 

with coefl&cients in R. Conversely, if relation (9) holds for the 
abscissas of three points of inflexion, the latter are collinear. 
For, the line joining two of them, (xj, yj) and (xi, yt), meets 
the curve at a single point {x, y) and that point is a point of 
inflexion, so that \f/{x, Xj, a;t)=0 and X{x)=0 have a unique 
common solution x. Thus x = Xi and hence y = y<. 

Let a substitution of the group G of equation X for the 
domain R replace three roots Xi, Xj, Xt, for which (9) holds, 
by the roots Xr, Xs, Xt. By property B ( 149) of the group 
G, 4^(xr, Xs, Xt) = 0. Hence every substitution of G replaces the 
abscissas of three collinear points of inflexion by the abscissas of 
three collinear points of inflexion. 

Denote by 1, 2, 3 the points of inflexion on one side of a 
triangle of inflexion; by 4, 5, 6 those on a second side. Those 
on the third side may be denoted by 7, 8, 9 in such an order 
that 1, 4, 7 are collinear; 1, 5, 9 collinear, and hence 1, 6, 8 
collinear. The third point on the line joining 2 and 4 is 8 or 
9 (since 4 and 7 are collinear with 1) ; if it be 8, we interchange 
symbols 5 and 6, 8 and 9, and see that all the earlier collineari- 
ties are preserved, and that 2, 4, 9 are now collinear. Then 
the point on the line joining 2 and 6 must be 7 (since not 8 or 
9), and that on the line joining 2 and 5 must be 8. We see that 
the 12 sets of collinear points of inflexion are those given by the 



176] 



GROUP FOR THE INFLEXION POINTS 



335 



rows, columns, and positive and negative terms of the expansion 
of the determinant 

1 4 7 

2 5 8 

3 6 9 

Henceforth we shall denote the abscissas of the nine points 
of inflexion by the nine symbols [^77], where | = 0, 1, 2 and 
77 = 0, 1, 2. Then the abscissas of collinear points of inflexion 
are those in the rows, columns, and positive and negative 
terms of 

[00] [01] [02] 

(10) [10] [11] [12] 

[20] [21] [22] 

and have the sum of their first indices divisible by 3, and also 
the sum of their second indices divisible by 3. 

Hence G is a subgroup of the group L of those substitutions 
on the nine roots which replace any three distinct roots [^jj], 
i = \, 2, 3, for which 

(11) 6+?2+f3=0, 771+172+173=0 (mod 3), 

by three distinct roots \^\ r}'i\ also satisfying congruences (11). 
We obtain a substitution of L if we take 



a b 
A B 



iO (mod 3), 



(12) ^'=ak-\-bv+c, v'=A^-\-Bv-\-C, 

where a, . . . , C are integers. For, 

3 3 3 3 

^^'i = a^^i-^b'^Vi-\-^c=0, '^v'i^O (mod 3). 

<-i <-i i-i -i 

Further, the [|'i 17 'J are distinct. For, if {'i = {'2, i7'i=ij'2, 
then 

aiii-^2)-\-b(m-V2)^0, A{^i-^2)+B(r,i-v2)^0 (mod 3). 
But the determinant aB-bA is not congruent to zero. Hence 
^1 = ^2, 171 =i?2, contrary to hypothesis. 



336 INFLEXION POINTS OF A CUBIC CURVE [Ch.XVIU 

Conversely, every -substitution 5 of L is induced by a linear 
transformation (12) on the indices. Let 5 replace [00] by 
[c C] ; the same is true of the substitution P induced by 

(13) r^^+c, 77'^77+C (mod 3). 

Hence S = TP, where T is a substitution of L which leaves 
[00] unaltered. Let T replace [10] by [aA], where therefore 
a and A are not both zero. Thus we can find two integers 
b and B such that aB bA is not divisible by 3; if a^^O, we 
may take 5 = 1, J = 0; if -4?^0, we may take & = 1, 5 = 0. Then 
the substitution P' induced by 

(14) J' = a^+H ri'=A^-hBr,, aB-bA^O (mod 3) 

replaces [10] by [aA]. Hence T=T'P', where T' is a substi- 
tution of L which alters neither [00] nor [10] and hence not 
[20], in view of the first column of (10). Let T' replace [01] 
by [de], so that gj^O. The substitution Pi induced by 

leaves unaltered each [^0] and replaces [01] by [de\. Hence 
T'Pi~^ leaves unaltered each [^0] and [01] and is the identity. 
For, it leaves fixed [02] by the first row of (10), and hence [21] 
and [12] by positive terms of the expansion of determinant 
(10), and then [1 1] and [22] by the second and third rows. Hence 
T'=Pi and S=PiP'P, so that S is induced by a linear trans- 
formation (12). 

Now [cC] was any one of 3X3 roots, [a A] any one of 3^1 
roots, and [de] any one of 3X2 roots. 

Theorem.* The group G oj the equation X Jor the abscissas 
{^7}] oj the nine points of inflexion is a subgroup of the group 
L of all of the 9X8X6 linear transformations (12) on ^, rj. 

The 3^ 1 incongruent linear homogeneous functions of ^ 
and 77 with integral coefl5cients modulo 3 are 

{, 17, i^+v), (-r?). 

* Jordan, TraiU des Substitutions, p. 302, where L is defined to be the group 
leaving (formally) unaltered the cubic function given by the sum of the products 
of the roots in each row, etc., of (10). But formal invariance may well intro- 
duce some confusion since the roots are not independent. For a wholly different 
determination of L, sec Weber's Algebra, 2d cd., vol. 2, p. 413. 



177] GROUP FOR THE INFLEXION POINTS 337 

Hence they are permuted by the linear homogeneous trans- 
formations (14), which form a group H. Since H permutes 
their four squares ^, . . . , (f t;)^, it is isomorphic with a 
group of substitutions on four letters. To show that the 
isomorphism is (2, 1), note that a transformation leaving ^ 
and rf unaltered at most changes the signs of | and tj. But 
if the sign of one is changed and the sign of the other is not 
changed, {^-{-nY is replaced by {k nY- Hence the identity / 
and 

J- k'=-^, v'^-v (mods) 

alone leave each of the four squares unaltered. 

The order of H is 48. There are 2X2X3 transformations 
(14) with b = 0, whence a^l or 2, B = l or 2, A=0, 1 or 2. 
There are 2X2X3^ transformations (14) with 6?^0, whence 
a and B are arbitrary, while aJ5 M = l then determines A. 

Hence * the quotient group H/{T, J} is simply isomorphic 
with the symmetric group on four letters. Thus H is a. solv- 
able group. 

The group T of the nine translations (13) is invariant under 
L. In fact, (14) transforms (13) into the translation 

^' = ^-\-ac-\-bC, v'=v+Ac-\-BC. 

It follows that L is a solvable group. By 68, Ex. 4, or by 
178, its subgroup G is solvable. Hence the equation X for the 
abscissas of the nine points of inflexion is solvable by radicals. 

177. Group of the Resolvent Quartic Equation (6). Let 
fi, . . . , f4 be the roots of (6) and set >'i=rir2 4-^3^4, etc., 
as in Example 2, 150. Then yi, y2, ys are the roots of 

f - Qbf + 1 2b^y - 16a2 - 721^ = 0. 

Setting y = z-{-2b, we obtairi the reduced cubic z^=D, where 

D = lQ{a^+Al^)j^O, 

by (7). From f its discriminant, we see that 



p = (yi-y2)(yi-y.,i){y2-y:i)=DV-27. 

* Another proof follows from the fact that the linear fractional transformation 
on s= ^/fi, derived from (14) by division, permutes the values 0, 1, 2, oo of s. 
t Or from (s w2)(s w*z)(w2 *), where w^+v+lfO, 



338 INFLEXION POINTS OF A CUBIC CURVE [Oh. XVIII 

Consider the special cubic form given by F in 175 with 
= 1, h= \. Then Z)= 48, so that P and each yi is irra- 
tional. Equation (6) is now 

(6') f4-6r2-f4r-3 = 

and is irreducible in the domain R{\) of rational numbers. 
For, no root is 1, 3, so that no root is rational. Further, 
a yi occurs in the coefficients of any quadratic factor, as shown 
by Ferrari's method of solving quartic equations. Hence 
the group of (6') for the domain R{\) is the symmetric group 
(Ex. 5, 153). 

Let/ be a cubic form whose ten coefficients are independent 
variables. Let R be the domain of the rational functions with 
rational coefficients of these ten variables. Then (Ex. 2, 
153), the group of the quartic equation (6) for R is the symmetric 
group. 

178. Group G of Equation X is the Linear Group L. After 
the adjunction of a root r of the resolvent quartic (6), the 
product of the equations of the three sides of an inflexion tri- 
angle has its coefficients in the domain {R, r), and the group G 
reduces to the subgroup which permutes the triples of abscissas 
of the points of inflexion on the sides of that triangle. First, 
let the triangle be that one whose sides contain the points of 
the rows in (10); these triples are merely permuted when the 
sign of either index is changed and also by the transformation 

and hence by the group of the 4-3^ transformations (12) with 
b=0, whose index under L is 4, By the interchange of the two 
indices, these triples are replaced by those in the columns of 
(10), so that the latter are merely permuted by the group 
of the transformations (12) with A=0. When b^A=0, we 
have 

^' = a^+c, n^Brj+C, aB^O (mod 3). 

Unless a = B, the triples in the positive terms of the determinant 
(10) are replaced by those in the negative terms, since this is 
true for $' = ^, v''n, and since each transformation 
(15) ?' = ?+:, v'^v-\-C 



178) GROUP FOR THE INFLEXION POINTS 339 

permutes the three triples in the rows, the three in the columns, 
etc., of (10). Hence after the adjunction of the four roots of (6), 
the group G reduces to a subgroup 2 of the group of the 18 trans- 
formations (15). 

The group 2 is of order a multiple of 2. Take a = 2, h = Q 
in 175. Then (6) becomes r(r^ 8) =0, all of whose roots are 
in /?() =R{y/^Z). The Hessian ( 175) is 

fl'=-24a;2(x32+6xi2). 

After the adjunction of the four r's, the domain is J?(V 3), 
to which does not belong the irrationality * V^V 3 occurring 
in the sides of the inflexion triangle H = 0. 

The group 2 is of order a multiple of 3. For 

f=Xi^-\-2X2^-\-'iX3^-\-QXiX2X3, 

the Hessian is Q^h', where 

h'= o^i^ 2:^2^ 4a:3^ + 10a;iX2a;3. 

Then Zh' -\-rf has a Unear factor if r = 3, -1, -14V^. 
This is evident for r = 3. For r= 1, we get 

^h'-f=4.{-Xi^-2X2^-^Z^ + QXiX2Xz), 

having the factor 

^^X2 + ^XZ. 

V2 

After the adjunction of the four r's, the domain is i?(V 3), 
to which "V^ does not belong. Thus the order f of 2 is a 
multiple of 3. 

We return to a cubic form / with arbitrary coefficients 
and the domain R defined by them. By the adjunction of the 
nine roots of X, the group G24 of the resolvent (6) is reduced 
to the identity. In fact, the abscissas and hence the ordinates 
of the nine inflexion points are in the enlarged domain. Thus 
the ratios of the coefficients of the equation of the line joining 

* No radical other than these two occurs in the sides of triangle (8) for r'^S. 
Hence the group G for this special cubic curve is of order 4. 
t Its order is in fact exactly 3; that of G is 6. 



340 INFLEXION POINTS OF A CUBIC CURVE [Ch. XVIU 

three collinear inflexion points are in the enlarged domain; 
the same is true for their products by threes giving the inflexion 
triangles h-\-2irf, so that each root r of (6) is in that domain. 
It now follows from the theorem of Jordan ( 167) that the 
adjunction to R of the four roots of (6) reduces the group G 
of X for R to an invariant subgroup 2 of index 24 under G, 
such that G/2 is simply isomorphic with G24. 

This group 2 was shown to contain a transformation (15) 
of period 3, necessarily a translation (13). By interchanging 
^ and rj if necessary, we may assume that ^ is. altered. Then 
the translation or its square is of the form 

Introducing ^ and 7; /^ as new variables, we obtain a group 
2i conjugate with 2 under L and containing the translation 
^' = 1+1, r}' = ri. The only transformations (12) which are 
commutative with this one are those with a = l, ^=0, Bf^O, 
b, c, C arbitrary, 2-3^ in number. The above translation is 
transformed into its inverse by ^'= ^, v'v- Hence exactly 
4 3^ transformations of L transform into itself the cycUc group 
of order 3 generated by it. Since this number is one-fourth 
of the order of L, a subgroup of index 3 under L cannot trans- 
form this cyclic group into itself. 

But 2 is of order 6 or 18. In the first case, 2 contains 
a single cycUc group of order 3, which is therefore invariant 
under G; while G is of order 24-6 and hence of index 3 under L. 
Thus the first case is excluded by the preceding result. Hence 
2 is of order 18 and G = L. 

Theorem.* If the coefficients of a cubic curve /=0 are 
independent variables, the group of the equation upon which 
depends the nine points of inflexion [^77], J, rj = 0, 1, 2, for the 
domain of the coefficients, is the group of all linear transformations 
on ^ and t? modulo 3. 

After the adjunction of the roots of the resolvent quartic 
(6), the group is that of the 18 transformations (15). The 

* Stated, but not completely proved, by Weber, Algebra, ed. 2, vol. 2. pp. 
416-7. The proof is due to Dickson, Annals of Math., ser. 2, vol. 16 (1914), pp. 
50-66. 



1791 REAL POINTS OF INFLEXION 341 

product P=A+24r/ of the linear functions which vanish at 
the sides of an inflexion triangle has as its coefficients quanti- 
ties in the enlarged domain. The determination of the linear 
factors requires the solution of a cubic equation. Consider the 
inflexion triangle associated with the rows in (10); after the 
adjunction of the roots of the corresponding cubic equation, 
the group permutes the roots [^17] in the same row. The only 
transformations (15) having this property are ^' = ^, ri'=rj-\-C, 
which form a group C3. The group of the resolvent cubic is 
therefore of order f -18 = 6. In the new domain, the group of 
the corresponding resolvent cubic for another inflexion triangle 
is C3. After the adjunction of one and hence all of its roots, 
we have the sides of two inflexion triangles, and their inter- 
sections give the nine inflexion points. 

Hence the determination of the inflexion points of an arbitrary 
cubic curve requires the extraction of a cube root and three square 
roots to solve the resolvent quartic equation, then the extraction 
of a square root and two cube roots to solve the two cubic equations 
which determine the sides of two inflexion triangles. IVo one of 
these three cube roots and four square roots can be avoided or ex- 
pressed rationally in terms of the others. 

179. Real Points of Inflexion. Let the coefficients of 
the equation of the cubic curve be real. After a suitable choice 
of axes, the nine abscissas of the points of inflexion are the nine 
distinct roots of an equation with real coefficients ( 176). 
Hence at least one point of inflexion is real. The reduction to 
the form F in 175 can therefore be effected by a real trans- 
formation. By 177 the discriminant of the real quartic 
equation (6) is 21D^ and hence is negative. Thus * there 
are two distinct real and two imaginary roots. One of the real 
roots is positive and the other is negative, as shown by the 
values 00 , 0, -}- 00 of the variable r. By use of the same 
values we see that the slope of the curve y= a:* -1-66/^-1- . . . , 
corresponding to (6), is positive at the point whose abscissa 
is the positive root and negative at that with the negative root. 
At the points of inflexion (1, r, 5) the slope is -As^, by the 

* Dickson's Elem^nlary Theory of Equations, p. 45, or Ex. 5, p. 101. 



342 INFLEXION POINTS OF A CUBIC CURVE [Ch. XVIII 

formula below (6). To obtain a real point, we must therefore 
take the negative real root r. Thus a real cubic curve F = 
has exactly three real points of inflexion, viz., (0, 0, 1) and 
(1, r, 5), where r is the single negative root. Any real cubic 
curve without a double point has exactly three real points of 
inflexion. 



CHAPTER XIX 

THE TWENTY-SEVEN STRAIGHT LINES ON A GENERAL CUBIC 
SURFACE AND THE TWENTY-EIGHT BITANGENTS TO A 
GENERAL QUARTIC CURVE 

180. Existence of the 27 Lines. We shall first show that 
there is at least one real or imaginary straight Une 

(1) x = mz-{-n, y = pz-\-q 

on the general cubic surface (f){x, y, z)=0. Eliminating x and 
y, and equating to zero the coefficients of the resulting cubic 
function of z, we obtain four relations between the four param- 
eters m, n, p, q. These are consistent and have one or more 
sets of solutions, except possibly for special sets of coefficients 
of </). In fact, they are evidently consistent when <f>=xyz. 
See also 183. 

To determine the number of the straight lines on the cubic 
surface, we employ homogeneous coordinates, choosing the 
tetrahedron of reference so that 0:3 = 0, X4 = are the equa- 
tions of a line on the surface. Then no one of the terms xi^, 
0:1^0:2, 0:10:2^, X2^ occurs in the equation of the surface, which 
is therefore of the form 

X3f-{-X4g = 0, 

where/ and g are homogeneous quadratic functions of Xi, . . . , 
X4. Part of the intersection of the surface by the plane X4 =CX3 
is the line X3 = X4 = 0, and the remaining part is the conic 
/i+cgi=0 in that plane, where /i and gi are derived f rem / 
and g by replacing x^ by cxs. Hence in/i and gi, the coefficients 
of Xi^ are quadratic in c, the coefficients of XiXi and 0:2X3 are 
linear in c, and the coefficients of xi^, X1X2, xj^ are free of c. 
The conic degenerates into a pair of straight lines if and only 

343 



344 LINES ON A CUBIC SURFACE [Ch. XIX 

if the Hessian (discriminant) oi fi+cgi is zero. The degrees 
in c of the second partial derivatives of /i -\-cgi exceed by unity 
those of gi. Hence the degrees in c of the elements of the Hes- 
sian determinant are 

1 1 2 

1 1 2 

2 2 3. 

Thus the determinant is of the fifth degree in c. There are 
cubic surfaces S for which this quintic has five distinct roots, 
so that the surface contains five pairs of straight lines intersect- 
ing the line C given by X3=Xi = 0. This is true if 

f = XiX2+eXiX4-{-tX2X^-\-aX3Xi-\-bX4^, g=Xi^-^X2XA. 

For then 

fi-]-cgi=cxi^-{-xiX2-\-cexiX3-\-{c^-\-tc)x2X3 + iac+bc^)x3^, 

whose discriminant is 

-2c[c*+2t(:^~{e-t^)c^+{b-te)c-\-a]. 

The second factor becomes any assigned quartic function by 
choice of t, e, b, a. Consequently, after the exclusion of the 
special surfaces for which the quintic is identically zero in c 
(for example, xyz = 0), or has fewer than five distinct finite 
roots, there remain surfaces of type S. For such a surface, 
each root c leads to a pair of lines A and B forming with C a 
triangle, which is the complete intersection of its plane X4 = cx3 
with the cubic surface. 

The line C was any straight line on the surface. Hence 
every line on the surface is met by ten other straight lines 
lying on the surface. 

Any straight fine L on the surface meets the plane of the 
triangle ABC and meets it at a point on one of the sides, since 
the triangle is the complete intersection of its plane with the 
surface. Thus L is cither A, B, C, or one of the eight lines, 
other than B and C, which meet A , or one of the eight new lines 
meeting B, or one of the eight new lines meeting C. These 
24 new lines are distinct, since otherwise one of them would meet 



181] 



DOUBLE-SIX CONFIGURATION 



345 



two of the lines A, B, C, whereas the triangle ABC is the com- 
plete intersection of its plane with the surface. We readily 
exclude the case in which L passes through the intersection 
of A and B. For, if so, we may take those three lines as con- 
current edges of a tetrahedron of reference. Then xi = X2 = 0, 
:j^i=^3 = 0, X2 = :r3 = are lines on the surface, whose equation 
= therefore has no terms in X3 and X4 only, none in xo and X4 
only, and none in xi and X4 only. Thus x^ occurs only in the 
terms Xia;2X4, XiX^Xi, 0:2^3^4. Hence the first partial deriva- 
tives of <^ with respect to each Xi vanish at (0, 0, 0, 1), which 
is therefore a singular point. But not every cubic surface has 
a singular point. Hence there are exactly 27 distinct straight 
lines on a general cubic surface. 

181. Double-six Configuration. Consider a line h\ on the 
cubic surface and the five pairs a, c (? = 2, . . . , 6) of lines 




Fig. 16. 

on the surface which meet h\. The five planes hiatc^ are dis- 
tinct and three Unes on the surface do not concur ( 180). 
Hence no two of the lines C2, . . . , ce intersect. The locus of 
a line L intersecting C2, C3, C4 is a surface of the second order * 

* By choice of the axes, the equations of ci become y=mx, s=c, and those 

of Cj become y= mx, z=c. The line L joining the general point (a, ma, c) 

of C2 with the general point (6, mb, c) of cj is 

ab a+b 

xdz-\-ce, y = tnez+mcd, dm , e= . 

2c 2c 

This meets Cz: x=h+t, y=rz-\-s, if 

idl)z+cet=0, (mer)z+mcds=0. 

In the determinant of the coefficients of z and the constants, replace d and by 
the values obtained by solving the equations of L. We get 

mxzcylmZ, cyzmc*xnUZ 



A= 



yzmcxrZ, mcxzc*ysZ 



'0, 



346 LINES ON A CUBIC SURFACE (Ch. XIX 

and hence is an hyperboloid // of one sheet. By suitable choice 
of the axes, the equation of // is 

or tuvw = 0, if the binomials are designated /, etc. Hence 

\u = rw, { u = sv, 
\v = rt, \w = st 

are intersecting lines lying on H. When r varies, we obtain 
one set of generators; when 5 varies, we obtain a second set. 
Through each point of H passes one and only one generator of 
each set. We may assume that C2, cs, Ca belong to the first set 
of generators. Let Cs cut H at Pi and P2. Through P< passes 
one generator of the second set, which therefore meets C2, C3, 
C4, as well as C5. Hence C2, C3, d, C5 have two non-intersecting 
transversals, one of which is 61; call the second ai. Since ax 
contains four points of the surface (the intersections of a\ 
with C2, C3, C4, C5), it lies wholly on the surface. The line ai 
therefore meets one of the sides of triangle h\aQC&. As observed 
above, ai does not meet bi. Denote by cq that one of the 
remaining two sides which meets ai, and by ae the other side. 
Then no two of the lines ai, . . . , ao intersect. 

The line ai is met by five lines 62, . ^ . , fte of the surface 
besides C2, . . . , cq. No two of the lines h\, . . . , h& inter- 
sect. 

Now 62 meets one side of triangle hiazc^ other than C3, since 
cz meets the side a\ of triangle 0162^2- Hence 62 meets a^. 

The name double-six is given to this configuration of 12 lines 
Qt, bi {i = l, . . . , 6) such that no two a's meet, no two 6's 
meet, a< and bt do not meet, while at and bj meet if i^^j. 

182. The 45 Triangles on the Cubic Surface. Since each 
line on the surface is a side of five triangles ( 180) and each 
triangle has three sides, there are 5-27/3 or 45 triangles. 

where Z=z*cK Suppressing the terms involving Z in A, we get 

c(mxz :y)* c{yz mcx)^= c{m^x^ y-)Z. 
Hence A is the product of Z by the required equation of degree 2. 



183] GROUP FOR THE 27 LINES 347 

Write cu=Cii for the c (j = 2, . . . , 6) in 181. Then 
aibiCii and OtbiCn are triangles on the surface. Let C23 be the 
third side of the triangle determined by a2 and 63. Hence C23 
is distinct from each at and bt; also from Ci, since C2 is the only 
d meeting 02, and since C2 meets the side ai of aiftacs and hence 
does not meet 63. This new line C23 is thus not one of the ten 
lines meeting bi; nor does it meet cs, which intersects the 
side Z>3 of 0263^23. Hence C23 meets the side as of a3C3&i. Sim- 
ilarly, C23 meets &2- For like reasons, we may write Cu=Cfi 
for the third side of the triangle determined by a< and bj for 

,y=i, . . . , 6; vy. 

We now have notations a<, bt, ctj for the 27 lines on the 
surface, and have the 30 triangles atbjCij. 

Next, if i, j, k are distinct, Ct] and <: do not meet. For, 
if they met, their plane would contain Oi and 6, whereas the 
latter do not intersect. Finally, two c's having four distinct 
subscripts intersect. For example, C34 meets one side of 0261^12, 
but does not meet 61 or a2, since 02 meets the side 63 of a\b2CzA\ 
hence C34 meets C12. 

Thus the sides of the remaining 15 triangles on the surface 
are c's with the following sets of subscripts: 

12 34 56, 13 24 56, 14 23 56, 15 23 46, 16 23 45, 

12 35 46, 13 25 46, 14 25 36, 15 24 36, 16 24 35, 

12 36 45, 13 26 45, 14 26 35, 15 26 34, 16 25 34. 

183. The Group for the Problem. Let R be the domain 
defined by the ratios of the coefficients of the equation for the 
cubic surface. After a suitable choice of axes of coordinates, 
we may assume that no two of the 27 lines (l) on the surface 
have the same parameter q. By eliminating m, n, p between 
the four relations mentioned in 180, we obtain an equation * 
E{q) =0 of degree 27 in q with coefficients in R. Since n may be 
eliminated last, it is a rational function of q with coefficients 
in R. The same is true of m and p. Thus the coefficients of 
the equation of a line on the surface are rational functions with 

*It may be assumed to have no multiple root ( 145 first foot-oote). 



348 



LINES ON A CUBIC SURFACE 



[Ch. XIX 



1 





m 


n 





1 


P 


Q 


1 





mi 


fix 





1 


Pi 


qi 



coefficients in R of the corresponding root q of E{q)=0. 
Hence we seek the group G oi E{q)=0 for R. 
Line (1) is in a plane with another line 

(2) x = miz-\-ni, y = piz-\-qi 

on the surface, if the eight parameters satisfy the relation 
obtained by eliminating x, y, z between the four equations: 



= 0. 



In view of the remark that m, n, p are rational functions of q, 
this yields a rational relation f{q, q\) = with coefficients in R. 
Let this relation be satisfied. Since there is a single line (5-2) 
on the surface coplanar with two intersecting lines {q) and (gi), 

fiq, 92) =0, /(^i, g2)=0 

have a single solution 92. The next to the last step in the 
eb'mination of q2 gives ^2 expressed as a rational function of 
^.and q\ with coefficients in R, say q2 = r{q, qi). 

Since any substitution of G must leave q2 r{q, qi) unaltered, 
it must replace the three g's corresponding to three Unes of a 
triangle on the surface by three q's of another triangle. Hence- 
forth we shall denote the 27 ^'s by the 27 letters a<, bj, Cy of 
182. Thus G is a subgroup of the group F on the a<, bj, c^ 
which permutes the 45 triples having the same notation as the 
45 triangles in 182. 

We readily determine the order and generators * of T. 

First, 

J = {aibi) . . . (aebe) 

interchanges the triples A^^aiVwa-nd A^, and leaves unchanged 
the 15 triples at the end of 182. Thus J is in T. 

* Much simpler and more natural than the generators used by Jordan, Traill 
desSubsliliitions, p. ^17. 



183] GROUP FOR THE 27 LINES 349 

Next, the 45 triples are evidently permuted by the substi- 
tution [ij] on the 27 letters which is induced by the interchange 
of the subscripts i and j. For example, 

[12] = {aia2){bib2)(Ci3C23){CuC24){Cl5C25){Cl6C26)' 

Finally, the group r contains 

A = {aia3C24)ia2CliCZ4){a'iCl2C23)(b3blC56)(bsC.i(iCi6){b6C35Cl5), 

which permutes the 45 triples as follows : 

(Ai2A32A42)(Ai4A34A24)(Ai3 A31 C13C24C56) 
(A43 A21 Ci4C23C56)(A23 A41 <: 12^34^56)^^, 

where 

P=VAi5 A36 Ci6C24C35)(A25 C14C25C36 C1QC25C34) 
(A51 A50 A53)(A45 C12C36C45 C16C23C45), 

while Q is derived from P by interchanging the subscripts 5 
and 6. 

From these substitutions of T we evidently can derive one 
which replaces ai by any one of the 27 letters. If a substitu- 
tion of r does not alter ai, it must permute amongst themselves 
the pairs bj, cij{j = 2, . . . , 6) occurring in triples with ai. 
From 

B = (a5a3C24) (^2^45^34) (^4^25^23) (bsbscie) (61C36C56) (66C13C15), 

which is the transform of A by [15], and the [ij], we readily 
derive a substitution which leaves ai fixed and replaces 62 by 
any one of the ten letters bj, Cij {j>l). 

If a substitution of r leaves ai and 62 fixed, it leaves C12 
fixed and replaces 63 by one of the eight letters bj, cij(J = Z, 
. . . ,6). Such a substitution can be derived from the [ij], 
i,j = Z, . . . , 6, and B. 

If a substitution of T leaves fixed ai, 62, C12, 63, it leaves 
fixed ci3 and permutes the pairs a, Ci2(i = S, . . . , 6) occur- 
ring in triples with ^2, and permutes the pairs at, Cts (' = 2, 4, 
5, 6) occurring in triples with bs. Hence it permutes the letters 
C23, di, CLb, o,Q common to the two sets of pairs. Now [15][34] 
transforms A into 

(a6a4a23) (^2^35^34) (^3^25^24) (64&5C10) (61^40^50) (6o<^i4Ci8), 



350 LINES ON A CUBIC SURFACE [Ch. XIX 

from which and the [y], ij = ^, 5, 6, we get a substitution leaving 
fixed ai, &2, Ci2, bs and Cis and replacing C23 by any one of the 
four letters C23, ^4, as, da- Next, the [ij], i,j = A, 5, 6, leave 
also C23 fixed and permute a4, a 5, a& in six ways. 
Finally, a substitution of V which leaves fixed 

oi, 04, as, ae, ^2, 63, C12, C13, C23 
is the identity. For, by the above two sets of four pairs, it 
leaves fixed 03, Ci2, (I2, C13 (i = 4, 5, 6). Then by the triples 
containing C12, Ci3 (i = 4), it leaves fixed cse, c^e, c^s- By the 
triples containing one of the last three c's and C23, it leaves 
fixed Ci4, Ci5, C16. Hence it leaves fixed the fifteen c's and the 
six a's and consequently the six 6's, in view of the triples OibjCtj. 

The group T is generated by A, J and the [ij], i,j = l, . . . ,6 
and is of order 27 10 8 24 = 51 ,840. 

It was noticed that J gives rise to an odd substitution on 
the 45 triples. Hence r has an invariant subgroup // of index 
2 and order 25,920, composed of those substitutions of T which 
give rise to even substitutions on the triples. Various proofs * 
have been given of the simplicity of H. 

We shall next discuss the remarkable relation between 
the 27 lines on a general cubic surface and the 28 bitangents 
to a general quartic curve. From the group of the latter 
problem, we shall be able to conclude ( 189) that the group G 
of the former is identical with r. Thus G has the factors of com- 
position 2 and 25,920. In particular, the equation E{q) = 
for the 27 lines is not solvable by radicals. The equation has 
a resolvent of degree 45, corresponding to the 45 triangles, and 
a resolvent of degree 36, corresponding to the 36 double-sixes. 
But it has no resolvent of degree less than 27 other than the 
quadratic the adjunction of whose roots reduces the group to 
the subgroup // of index 2; this fact has been proved in three 
distinct ways.f The problem is thus of special complexity 
on the algebraic side. 

* Jordan, Trail4 des Substitutions, 444, 504; Dickson, Linear Groups, 
p. 307. 

t Jordan, TraitS des Substitutions, pp. 319-329; Dickson, Trans. Amer. Math. 
Soc, vol. 5 (1904), p. 126; cf. vol. 6 (1905), p. 48; Mitchell, ibid., vol. 15 (1914), 
p. 379. 



184] CUBIC SURFACES AND QUARTIC CURVES 361 

184. Relation between Cubic Surfaces and Plane Quartic 

Curves.* Using homogeneous coordinates, let 

f{x)=f{Xi, X2, Xs, X4)=0 

be the equation of a cubic surface without a singular point, 
so that the first partial derivatives of/ with respect to xi, . . . , 
Xi are not all zero for a set of x's not all zero. The point iy-\-\z) 
on the line joining the points {y) = {yi, . . . , y4) and (2) is 
on the surface if 

f(y-\-\z)^f(y)+\L-\-h\^Q-[-\%z) =0, 
where, by Taylor's theorem, 

L = 2i^+ . . . +24^, Q = z,^^^+2z^Z2: ^^ ^ 



dyi dyi dyi^ dyidy2 

Take (y) to be a fixed point P on the surface /=0. Then 
the line joining P and (2) meets the surface in two further 
points which coincide if 

(3) {lQy = Lf(z). 

Hence this equation of degree four in zi, . . . , 24 represents 
the tangent cone T4 to the surface /=0 with the vertex P on 

Any plane through a straight line / on the cubic surface 
meets the surface in I and a conic c. Let / and /' be the two 
intersections of / and c. The tangent to c at / is the Umit- 
ing position of a Une meeting the surface in three points and 
hence meets the surface at three coincident points. Thus 
/ and this tangent line are principal tangent lines to the cubic 
surface and their plane is a tangent plane. Hence any plane 
through I is tangent to the surface at two points /, /', and thus 
is a bitangent plane. 

In particular, the plane through the fixed point P and any 
line / on the cubic surface is a bitangent plane to the surface 
and hence also to the tangent cone T4' 

The section of T^ by an arbitrary plane is a quartic curve 
C4. The intersections of E with the above 27 bitangent planes 
to Ti give 27 bitangent lines to C4. 

* Geiser, M athematische Annden, vol. 1 (1869), p. 129. 



362 LINES ON A CUBIC SURFACE [Ch. XIX 

But there is an additional bitangent to d, making 28 in all. 
In fact, the plane L = passes through (y), in view of Euler's 
theorem (4), 173. Hence L = intersects T^, given by (3), 
in two pairs of coincident lines, the intersections of L = 0, Q = 0. 
Thus L = is a bitangent plane to T^. 

Before we can conclude that the general plane quartic 
curve has exactly 28 bitangents, we must show that, conversely, 
any given quartic curve C4 is the intersection of the plane 
U of the curve with the tangent cone to a suitably chosen cubic 
surface / at a point P on it. 

Let X, y, z be the homogeneous coordinates of a point in the 
plane V referred to a triangle of reference whose side s = is 

0=0 




Fig. 17. 

one of the bitangents to C4 and whose sides it; = and 3' = are 
any lines through the points of contact of this bitangent. Then 
the equation of C4 reduces to o^'f- = when z = and hence is 

(4) z<t>-x^y^ = 0, 
where </> is a cubic form in x, y, z. 

Let P be any point not in the plane U of C4. As the tetra- 
hedron of reference for homogeneous coordinates x, y, z, u of 
points in space, take that determined by the plane U and the 
planes through P and the sides of our triangle of reference. 
Then the desired cubic surface is 

(5) f=<t>+4uxy-\-AuH = 0. 

In fact, we shall proceed with this / as we did with the 
general / and find its tangent cone with the vertex P = (0, 0, 
0, 1) in place of (y). Now 

dx dx '^' dy dy 

^=9^+4u2, ^ = 4A:y+82. 
92 82 du 



184] CUBIC SURFACES AND QUARTIC CURVES 353 

At the point P these derivatives vanish, with the exception of 
the third, which becomes 4. Hence L = Az. 

The second partial derivatives vanish at P, except 

Hence 

\Q = 2xy-\-^zu. 

Thus the tangent cone T^ from P is 

(2xy+42M)2 = 40/. 

Deleting the factor 4 and inserting the value of /, we get the 
equation (4) of C4 in the plane m = 0. 

EXERCISES 

1. In the equation of the tangent plane to ^=0 at {U, . . . , /), the 
coeflScient of 2zi is 



9yi* dyidyj 9yi9y4 

which, for (/) = (y), equals 



'^^'a>'iVay./ 



9yi 



Hence the tangent plane to ^ = at {y) is =0. 

2. Let h, It, h be three lines on the cubic surface / which form a tri- 
angle and let P be a point of / not on one of the 27 lines. For i= 1, 2, 3, 
the plane Pit meets / in li and a conic ct. Call rj and 5 the points of 
intersection of U and ct; they are on T* and / and hence on ^=0. Thus 
the r<, Si are six points on the conic c in which Q=0 is met by the plane 
of /i, k, h- This plane meets the princii)al tangents /, ktof at P (the lines 
of intersection oi L=0, Q=0) at two points on c. 

Any plane cuts r in a quartic curve d. From A, /i, /j, we obtain 
bitangents /'i, I't, I' 3 to G whose points of contact are the projections 
of ft, Si (t=l, 2, 3) from P. From /i, /j, we obtain a bitangent t. The 
conic c projects into a conic c'. Hence C4 has the four bitangents I't, I'l, 
I't, T, whose eight points of contact are on a conic i'. 

3. Thus T and any one of the 4') triangles on / determine a conic tf. 
But any one of the 28 bitangents can be used in place of r. Since any 
one of the conies is related in this way to each of the four bitangents, there 



354 BITANGENTS TO A QUARTIC CURVE [Ch. XIX 

are 28-45/4 = 315 conies each containing the eight points of contact of 
four bitangents. 

4. If (x, y, z) is a singular point with Z9^0 on (4), then (5) has the sin- 
gular point (x, y, z, u), where 2zu+xy=(). 

185. Steiner Sets of Bitangents to a Quartic Curve. Let 
two sides, x = and >' = 0, of a triangle of reference for homo- 
geneous coordinates x, y, z be two bitangents to the quartic 
curve /=0. Then, for x = 0, f reduces to a perfect square, so 
that 

f=x<l>-\-{rz^-\-syz-\-ty^y, 

where <t> = c-^yq, c being a cubic form in x and z only, and q 
a quadratic form in x, y, z. Similarly, 

/v_o = {rz^-^uxz+vx^y. 

Hence the latter is identical with icc+r^z*. Thus 

f^xyq-\-{rz^-\-uxz+vx^yr^^+(rz^-\-syz-i-ty^y 

xyqi + {rz^+syz-\-ty^-\-uxz-\-vx^y, 
where 

qi =q 2{sz-{-ty){uz-\-vx). 

Hence /=0 can be given the form xyU=V^, where U and V 
are quadratic forms in x, y, z. Conversely, this equation 
evidently represents a quartic curve having x = and y = 
as bitangents. 

If X is any constant, the equation may be written as follows: 

xy{U-\-2\V-{-\^xy) = (F+Xa:^)^. 

Now U -\-2\V -\-\^xy is a quadratic form in x, y, z. Its 
discriminant (Hessian) is a determinant of the third order 
whose elements are linear in X with the exception of two ele- 
ments which involve X^. Hence the discriminant is of the 
fifth degree in X. It has no double root in general. For, if 

U = 2xz, V = ax^-\-bf-\-z^-\-2dxy-\-2exz, 

the discriminant is 

-|X[X*+8JX3+lG(J2-a6+5e2)x2-M66gX-{-4&], 

and the last factor can be identified with any quartic in X whose 
constant term is not zero. Hence if our quartic curve is general, 



185] STEINER SETS OF BITANQENT8 356 

there are five distinct values of X for which the quadratic form 
is a product of two linear functions. We thus have * 

Theorem 1. Given any two hitangents x = 0, y = to a gen- 
eral quartic curve, we can determine in exactly Jive ways a pair 
of lines ^ = 0, 77 = 0, such that the quartic becomes xy^rj = Q^. Then 
the eight points of contact of the four hitangents x, y, ^, t] lie on 
the conic Q = 0. 

Such a set of six pairs of hitangents is called a Steiner set 
(the older term was Steiner group). 

Let a,h] c, d; e,fhe three pairs of hitangents of the Steiner 
set determined by a, b. Then the quartic is abcd = Q^ or 

ab{cd-\-2\Q-^\^ab) = {Q-\-\aby. 

As above, we can determine X (Xp^O) so that 

cd+2\Q+\^ab=ef. 

Eliminating Q between this and abcd = Q^, we get 

^abcd = ('\ab-{-j-^A\ 

Replacing a, c, e by a/X, \c, \e, we get 

(6) Aabcd = {ab-\-cd-efY, 
which may be written in the symmetrical forms 

(7) a'^y^+cH'^-\-^f^ = 2abcd-V2abef-{-2cdef, 
(7') v^+V^+V^=0. 

Transposing the first radical, we derive 
4iCdef= {ef-\-cd aby. 

Hence the points of contact of c, d, e, f are on a conic. 

Theorem 2. The eight points of contact of any two pairs of 
bitangents of a Steiner set are on a conic. Thus tlie same Steiner 
set is determined by any one of its six pairs. 

Since the ^28 27 pairs of bitangents lie by sixes in a Steiner 
set, there are 63 Steiner sets. 

Also by 180 and Exs. 2, 3 of 184, with t and I'l as the given bitangents. 



366 BITANGENTS TO A QUARTIC CURVE [Ch. XiX 

Three bitangents are said to form a syzygetic or an asyzygetic 
triple according as their six points of contact are or are not on 
a conic. Thus a, b, c are syzygetic if they belong to a Steiner 
set and two of them, as a and b, form a pair of that set. We 
shall prove the important 

Theorem 3. Three bitangents a, c, e of a Steiner set are 
asyzygetic if no two of the three form a pair. 

First, a, c, e do not all pass through the same point P. For, 
if they did, P would be a singular point of the curve (6). We 
may therefore take them as the sides of a triangle of reference 
and write 

b = la-\-inc-\-ne, d = pa-\-qc+re, f=sa-\-tc-\-ue. 

No one of the constants /, q, u is zero. For, if / = 0, for 
example, the lines b, c, e would meet at a singular point of the 
curve (6). 

Suppose that the points of contact of a, c, e with (6) lie on 
a conic C=ga^-\-hc^+ke^-{- ... =0. Then, by (6) for a = 0, 

etc., 

Ca~o = 'K\c{qc-\-re) e{tc-\-ue)], 

Cc~o = fJiW{la-\-ne) e(sa-\-tie)\, 
CemO = v{'^{la-\-fnc)c(pa+qc)]j 
where X, n, v are constants. Hence 

h = \q, g = til, g = vl, 

k = \u, k=fxu, h=vq, 

whence \= v, fi = v, \ = n, or \ = fi = v = 0. Thus g = h = k = 
and the conic C = passes through each vertex of our triangle 
of reference ace. The vertex a = c = is not on /=0, since 
u^O. But the points of contact of a with (6) are given by 
cd ef=0 and, being on C = 0, are the vertices lying on a; so 
that the vertex a = c = is on /=0. The assumption that 
Theorem 3 is false has thus led to a contradiction. 

Corollary. A Steiner set a, b; c, d; . . . has in common 
with the Steiner set a, c; b, d] ... no bitangent except a, b, 
c, d. 



186] THE NOTATION OF HESSE AND CAYLEY 357 

For, if e is common to them, a, c, c are asyzygetic by the 
first, and syzygetic by the second set. 

186. The Notation of Hesse and Cayley. The 28 bitangents 
are designated [y] = [ii], i, j = \, . . . , 8; iji^j. We shall 
usually omit the brackets. We shall prove that the notation 
may be assigned to the bitangents in such a way that the 63 
Steiner sets are given by 

(8) agah, bgbh, eg ch, dg dh, eg eh, fgfh; 

(9) abed, acbd, ad be, ef gh, egfh, ehfg; 

where a, . . . , h form a permutation of 1, . . . , 8. For 
examples, see 55 and Si below. Since g and h may be any two 
of the eight figures, there are 8-7/2 or 28 sets of the first type. 
Since there are 70 ways of selecting four figures out of eight, 
there are 35 sets of the second type, which is determined by 
a, b, e, d or by e,f,g, h. 

Having in mind also another appUcation (191), we shall 
proceed abstractly and consider the distribution of 28 symbols 
into sets of six pairs, in which the arrangement of the pairs and 
the order of the symbols of a pair are immaterial, and such that 

A. Every pair occurs in one and but one set. 

B. If a, /S, and 7, 5 are two pairs of a set, then a, 7, and /3, 5 
are pairs of a set. 

C. Two sets a, /3; 7, 5; . . . and a, 7; /3, 5; . . . have 
only four symbols in common* 

For Steiner sets of bitangents, these properties hold in 
view of 185. 

As the first set we shall take 
Si. 12 34, 13 24, 14 23, 56 78, 57 68, 58 67. 
Rearranging the components of the first and fourth pairs by B, 
we see that there is a set S2 with 12 56, 34 78, and a set S3 
with 12 78, 34 56, which, by C, have no further symbol in 
common with each other or with 5"!. Hence they introduce 
all of the 8+8 symbols not found in Si. It is at our choice 
to select the eight to go into S2; we take 

52. 12 56, 15 26, 16 25, 34 78, 37 48, 38 47; 

53. 12 78, 17 28, 18 27, 34 56, 35 46, 36 45. 



358 BITANGENTS TO A QUARTIC CURVE [Ch. XIX 

The set Si with l2 57 and 34 68 has no further s3rmbol 
in common with Si and hence contains eight new symbols 
chosen from those of 5*2 and 53 not in the first or fourth pairs. 
The two of a pair cannot be two, as 26 and 25, from 52, since 
by B they would imply the pair 15 16, contrary to C. Again, 
if 25 occurs in 54, 16 does not occur; for, 25 jk and 16 /w would 
imply a set with 25 16 and jk Im, necessarily 52, whereas jk, 
Im are not in 52, by the last result. Hence after applying 
a substitution which permutes the eight symbols in 52 not in 
the first or fourth pairs, and the eight in 53 not in the first or 
fourth pairs, we may take 

54. 12 57, 15 27, 17 25, 34 68, 36 48, 38 46. 

The set 55 having 12 25 contains, in view of 52 and 54, 
the pairs 56 16 and 57 17, and no further symbols from 52 and 
54, and therefore six symbols chosen from 

13 24, 14 23, 58 67, 18, 35, 45, 28, 

the earlier ones being paired for the sake of simplicity of refer- 
ence. But 57 17, 28/ imply a set with 17 28, 57/, contrary 
to 53. Hence 28 is excluded. If we take at least four of the 
first six symbols, we must pair two of them with each other; 
these two occur in 5i and cannot be a pair in Si, so that we have 
a contradiction with C. Hence we must take one and but 
one from each pair 13 24, etc. Since we may interchange the 
symbols of any one of these three pairs without altering sets 
5i-54, we may assume that 55 contains 58, 13, 14. These 
and sets 5i-54 are not altered by 

(15 38) (26 47) (18 35) (27 46), 
(38 48) (47 37) (46 36) (35 45), 

which give rise to any permutation of 18, 35, 45. Hence we 
may take 

55.- 21 25, 31 35, 41 45, 61 65, 71 75, 81 85. 

Without the use of further substitutions to fix the choice of 
equivalent notations, we can now prove that the 63 sets are 



186) THE NOTATION OF HESSE AND CAYLEY 359 

uniquely determined by properties A, B, C and that each is 
of one of the two types (8), (9). 

Using 52, ^4, Si, Ss, Si in turn, and property B, we get 

Se. 15 48, 26 37, 27 36, 18 45, 14 58, 23 67; 

57. 15 38, 26 47, 27 46, 18 35, 13 58, 24 67; 

58. 37 47, 38 48, 36 46, 35 45, 13 14, 23 24. 

The set 59 with 12 68 has, by 5i, 54, the pair 34 57 and 
the symbols 16, 26, 47, 37 paired with 28, 18, 35, 45, since not 
paired with each other in view of 52. But 16 is not paired 
with one of the last three in view of 55. Hence S9 contains the 
pair 16 28. A pair 26 35 would imply 47 18 by 5? and hence 
37 45, contrary to Sg. A pair 26 45 would imply 37 18 by 56 
and hence 47 35, contrary to Sr- Hence, by 56, 

59. 12 68, 34 57, 16 28, 26 18, 37 45, 47 35. 

With 12 16 occurs 56 25 by 52, 55, and 68 28 by 59, while 
67, 24, 23 are paired with 27, 46, 36, by 5i. But 67, 23 are 
paired with 27, 36 by 56, and 67, 24 with 27, 46 by 57. Hence 

5io. 12 16, 56 25, 68 28, 67 27, 24 46, 23 36. 

By 52, 54, 53 we have the first four pairs of 

5ii. 26 16, 15 25, 17 27, 28 18, 13 23, 24 14, 

and four of the s>Tnbols 13, 14, 23, 24, 58, 67. Pairs 58 /, 28 18 
would imply 58 18, 28 1, contrary to 55. Pairs 67/, 17 27 would 
imply 67 27, tU, contrary to 5io. Finally, 13 14, 13 24 are 
excluded by Ss and Si. Hence we have Sn. By the same steps, 

5i2. 47 16, 38 25, 17 46, 28 35, 58 23, 67 14; 

5i3. 37 16, 48 25, 17 36, 28 45, 58 24, 67 13. 

By 57, 56, 5ii, 55 or 5io, Si, we get 

5i4. 26 27, 47 46, 37 36, 16 17, 56 57, 78 68; 

5i6. 15 18, 38 35, 48 45, 25 28, 56 68, 78 57. 

By 5i and 55, the set 5ifl with 12 58 has 34 67 and 25 18 
and the symbols 15, 47, 37, 28, 46, 36, since 26, 38, 48, 27 are 
excluded by 59, Su, S13, Sio, respectively (since 25 18, 26 imply 



360 BITANGENTS TO A QUARTIC CURVE [Ch. XIX 

that 25 is in a set Sq with 26 18). By 5i5, 15 is paired with 28 
in Si6. By Sg and Su, 47 is not paired with 37 or 46. Hence 

5i6. 12 58, 34 67, 25 18, 15 28, 47 36, 37 46. 
In exactly the same manner, we get 

5i7. 12 13, 34 24, 25 35, 38 28, 26 36, 37 27; 

Si8. 12 14, 34 23, 25 45, 48 28, 26 46, 47 27. 
By .Si, 5io, Si6 we get the first three pairs of 

5i9. 12 67, 34 58, 16 27, 17 26, 38 45, 48 35, 

and see that the further symbols are those in the last three pairs. 
By 5*4, 17 is not paired with 38 or 48, and by 53 not with 35 or 
45. Hence 17 is paired with 26. By Ss and S15, 38 is not paired 
with 48 or 35 and hence is paired with 45. 
Similarly, using 5*1, 6*10, S12, Sn, we get 

520. 12 24, 34 13, 16 46, 47 17, 15 45, 48 18, 

except as to the pairing of the last four symbols, which is deter- 
mined by So and S15. By Si, 5io, S13, Sis (and 5?, S15 for the 
pairing the final four symbols), we get 

521. 12 23, 34 14, 16 36, 37 17, 18 38, 15 35. 

We may now determine uniquely the remaining 42 sets 
from 5i-52i by use of the simple property B. Thus by 52, 
54, 521 and 5i6, 52o or 5i7, 5c, we get 

522. 12 15, 56 26, 57 27, 58 28, 23 35, 24 45; 

523. 12 38, 56 47, 57 46, 13 28, 23 18, 67 45. 

The remaining sets may be derived independently of each 
other from 5i, . . . , 523 by use of property B; they need not 
be tabulated here since they, like the earlier sets, are of one 
of the two types (8), (9). 

Theorem 4. There is a distribution of 28 symbols into 63 
sets of 6 pairs such that the arrangement of the pairs in a set is 
immaterial and such that properties A, B, C hold. Any su^h 
distribution can be derived from that given by (8) and (9) by a 
substitution on the 28 symbols. 



186) THE NOTATION OF HESSE AND CAYLEY 361 

Hence the notation ij may be assigned to the bi tangents in 
such a way that the Steiner sets are given by (8) and (9). 

Since the sets 5i, 52, Si together contain all the bitangents, 
any bitangent syzygetic with 12 and 34 occurs in S\ (Theorem 
3, 185). Similarly, every syzygetic triple is composed of three 
bitangents of a Steiner set, two of which form a pair. The 
converse was noted before Theorem 3. Any syzygetic triple 
of a Steiner set (8) is of the type ag, ah, bg; any one of a Steiner 
set (9) is of type ab, cd, ac or ab, cd, ef. The latter is repre- 
sented by 111, i.e., by three segments whose six end points are 
all distinct. The former is represented by the broken line 
n composed of the segments ba, ac, cd; and the same repre- 
sentation is given to ha, ag, gb. 

Theorem 5. A triple is syzygetic if and only if it is of type 
\\\ or type fl . Hence a triple is asyzygetic if and only if it is of 
one of tlie three types A, VI, \|/. 

Seven bitangents are said to form an Aronhold set if every 
triple contained in it is asyzygetic. An example is 

(10) 18, 28, 38, 48, 58, 68, 78, 

each triple of which is of type \\/ and hence asyzygetic. 

An Aronhold set is represented by a figure formed by 
seven segments with at most eight end points. It is not com- 
posed of more than two separate parts; for, if it be, it would 
contain three syzygetic bitangents |||. If a separate part has 
two or more points from each of which at least two segments 
radiate, that part is of type A, since otherwise it would contain 
a triple of type fl. Hence each separate part is a A or a fan 
Fn consisting of n segments radiating from one vertex. 

First, let there be a single part. Then the set is a fan F7. 
Its vertex may be any one of the points 1, . . . , 8, so that 
there are eight such Aronhold sets. That with the vertex 8 
is given by (10). 

Finally, let there be two parts. If they are fans Fr and F 
the number of segments r-\-s is 7, and hence the number of 
points is f -M +5-|- 1=9, whereas at most 8 points occur. Hence 
one part is a A and the other a fan F4 ; for example, 

(11) 12, 23, 31, 48, 58, 68, 78. 



362 BITANGENT8 TO A QUARTIC CURVE [Ch. XIX 

The triangle can be chosen in 56 ways (the number of ways of 
selecting three p)oints out of eight), and then the vertex of the 
fan is any one of the remaining five points. Hence (11) is one 
of 280 such sets. 

Theorem 6. There are exactly 288 Aronhold sets. Each 
is a Jan Fj or is composed oj a triangle and a Jan F\. 

187. Group of an Equation for the 28 Bitangents. Take as 
the ic-axis of Cartesian coordinates a line not through the 
intersection of any two bitangents. Then the bitangents 
cross the :r-axis at 28 distinct points (^, 0), and each bitangent 
y = m{x ^ is uniquely determined by its f. If the equation 
to the quartic curve is F{x, y)=0, then F[x, m{x ^\ must 
be a perfect square in x. From the resulting two conditions 
on m and ^, we obtain w as a rational function of ^ and the 
coefficients of F{x, y), since there is a single m for each ^; and 
hence obtain an equation E{^) =0 of degree 28 whose coefficients 
are in the domain R of the rational functions of the ratios of 
the coefficients of F. Let G be the group of this equation for 
the domain R. 

Let ^ = 0, q = be the two bitangents determined by the 
roots ^1, ^2. Then (185) there are two conies U = 0, V = 0, 
with coefficients in the domain (R, ^i, ^2), such thsit F=pqU V^. 
As there proved, there are five values of X for which 

U+2\V-{-\^pq=0 

is a pair of lines. Li the product 

P = n(C/+2XF+X2/>g), 

extended over these five values of X, the coefficients are in the 
domain (R, ^1, ^2). Now P is a product of ten linear functions 
of X and y, which represent ten bitangents forming with p and 
q a Steiner set. Settii\g y = in P = 0, we obtain an equation 
Ji^i, ^2, ^)=0, with coefficients in R, whose ten roots x are 
the ^'s of these ten bitangents. Thus/(^i, ^2, ^3) is zero when 
{1, h, h are roots of (^)=0 corresponding to a syzygetic 
triple of bitangents and not zero for three roots corresponding 
to an asyzygetic triple. In the first case, we shall call the roots 
syzygetic; in the second case, aszygetic. 



187] GROUP FOR THE BITANGENTS 363 

If a substitution 5 of G replaces three syzygetic roots {i, 
I2, ?3 by the roots ^'i, ^'2, ^'3, then, by property B of 149, 
/(^'i, $'2, ^'3)=0 and ^'1, ^'2, ^'3 are syzygetic. The converse 
is shown to be true by use of S~^ 

Theorem 7. Every substitution of G replaces syzygetic roots 
by syzygetic roots and replaces asyzygetic roots by asyzygetic roots. 

Corollary. Every substitution of G replaces any Steiner 
set by a Steiner set and every Aronhold set by an Aronhold set. 

We proceed to exhibit a fixed group r which contains G as 
a subgroup no matter how we vary the coefficients of the quar- 
tic equation and hence vary G. For the extreme case in which 
the coefficients are independent variables, we shall later see 
that G is identical with r. But in every case we shall in the 
meantime be able to determine the possible number of real 
bitangents from a knowledge of the substitutions of period 
2 in r. 

The group G may contain a substitution S which interchanges 
the roots 18 and 28 of the Aronhold set (10) and leaves unaltered 
the remaining roots i8, i = 3, . . . , 7. If so, 5 replaces the 
Steiner set la 8a(a = 2, . . . , 7) by a Steiner set 2 in which 
the figure 8 occurs six times and hence is of type (8) with A = 8 
(since we may permute g and h). Since 8 is paired in 2 with 
any figure except 2 and 8, we have g = 2. Hence 2 is 2a 8a 
(a = 1,3, . . . , 7), so that 5 replaces la by 2a if a > 2, and leaves 
12 unaltered. Next, 5 replaces S by a Steiner set of type 
(8) with /j = 8 and containing 21 82, and hence with g = l, 
so that the new set is 21 82, la8a (a = 3, . . . , 7). Thus 
5 replaces 2a by la if a>2. Finally, 5 replaces the Steiner 
set 31 81, 32 82, 3a 8a (a=4, . . . , 7) by one with 32 82, 
31 81, and hence with 3a 8a, so that S leaves unaltered 3a 
(a = 4, . . . , 7). Similarly, S leaves unaltered ba {b, a = 3, 
. . . , 7; b9^a). Thus S is the product of the transpositions 
(ly, 2y),y = 3, . . . , 8, and is therefore the substitution on the 
28 symbols [ij] which is induced by the transposition of the 
two indices 1 and 2. 

We obtain in this way 7! substitutions which leave unaltered 
the Aronhold set (10), merely permuting its seven symbols. 



364 BITANGENTS TO A QUARTIC CURVE [Ch. XIX 

The group G may 'contain a substitution 5 which replaces 

the symbols (10) by the corresponding symbols of the Aron- 

hold set 

17, 27, 37, 47, 57, 67, 87. 

If so, S replaces the Steiner set ia 8a (a = 1, . . . , 7; a^i), 
where i is a fixed integer = 6, by a Steiner set of type (8) with 
/f = 7, g = i, since 7 is paired with every integer except 7 and i. 
Hence the new set is ia7a (a = l, . . . , 6, 8; aj^i). Hence 
5 leaves fixed ia (a ^ 6) and replaces i7 by i8. Thus S is induced 
by the transposition of the indices 7, 8. 

We now have 8! substitutions which permute the eight 
Aronhold sets typified by fans F7 with vertices 1, . . . , 8. 
These 8! substitutions on the 28 symbols are those induced 
by the 8! substitutions on the indices 1, . . . , 8, and form a 
group E. 

The group G may contain the substitution 

^/18 28 38 48 58 68 78 ... \ 
^^^^ \23 13 12 48 58 68 78 . . . /' 

which replaces the Aronhold set (10) by the Aronhold set (11) 
composed of the triangle with vertices 1, 2, 3 and a fan Fi 
with the vertex 8. Since P replaces the Steiner set la 8a (a = 2, 
. . . , 7) by one of type (8) with h = S and having the symbols 
13, 12 and hence with g = l, P replaces 12 by 38, 13 by 28, 
and leaves unaltered la(a = 4, . . . , 7). Next P replaces the 
Steiner set 8a 4a (a = 1, 2, 3, 5, 6, 7) by that determined by 
23 14 and hence leaves 24 and 34 unaltered and replaces 45 
by 67, 46 by 57, 47 by 56. In this way we find that 

Pi238 = (12 38) (13 28) (23 18) (45 67) (46 57) (47 56), 

which may therefore be designated also by P45G7. 

Similarly, or by symmetry, we obtain the substitution 

i'aia,aia4 = -Pftft/SW5 = (l2 a3a4)(aia3 OC2a4,) 

(aia4 a2a3)(i3i/32 fi3^A)(fii^3 m^Wi^A ^2^3), 

where ai, . . . , /34 form a permutation of 1, . . . , 8. There 
are 35 such substitutions, since there are 70 combinations of 8 
things 4 at a time. 



188] NUMBER OF REAL BITANGENT8 366 

We thus have a group, of order 288- 7! =36-8!, 

188. Number of Real Bitangents. We shall employ the 
Lemma.* // an algebraic equation with distinct roots and 
with coefficients in a real domain R has exactly v pairs of con- 
jugate imaginary roots X2j-i, X2j(j = l, . . . , v), its group for 
R contains the substitution 

S={XiX2) . . . (x^j-iX^j) . . . (^2,-1X2,). 

If we apply the Corollary in 149 to the group |1, S\, we 
see that our Lemma is proved if we show that S leaves numeri- 
cally unaltered every rational function <l>{xi, . . . , Xn) oi the 
roots such that 4> has its coeflEicients in R and equals a quantity 
in R. Since the numerical value of <t> is real, remains numeri- 
cally unaltered when i = A/^ is changed into i in. each im- 
aginary root; but the resulting change in <j) is the same as if 
we had applied the substitution S. 

Corollary. Either the roots are all real or else the number 
of real roots equals the fiumber of letters mialtered by one of the 
substitutions of period 2 of the group for R of the equation. 

We shall prove that every substitution S of period 2 of the 
group r of 187 leaves fixed exactly 4, 8 or 16 of the 28 sym- 
bols. First, if 5 is in the subgroup E, it is induced by 1 , 2, 3 
or 4 transpositions on the 8 figures and hence leaves unaltered f 
16, 8, 4 or 4 symbols, respectively. Second, S = Pi234 leaves 
16 symbols unaltered. Third, let 5 = a"^Pi234, where a is not 
the identity and is induced by the substitution which replaces 
1, . . . , 8 by ai, . . . , ag. Then 

5=5~^=-Pl234 <T = (rPaiatctuu ' Pl234=<rS = O^Pa^ata**- 

Hence (end of 187) 0^ is the identity and ai, a2, 03, a^ form a 
permutation of either 1, 2, 3, 4 or of 5, 6, 7, 8. In the latter 
case, <T is induced by 

(1 ai) (2 aa) (3 ag) (4 ^4), 

* A less precise theorem limited to irreducible equations was given by E. 
Maillet, Annates de Toulouse, ser. 2, vol. 6 (1904), p. 280. The present Lemma 
was given by Dickson, Annals of Matkemalics, ser. 2, vol. (190.')), p. 144. 

fThe verification can be made for the substitutions used below. 



366 BITANQENT8 TO A QUARTIC CURVE [Ch. XIX 

SO that the substitution induced by 

(ai a2 as aA 
5 6 7 8/ 

transforms 5 = aPi234 into that induced by (15) (26)(37)(48)Pi234, 
which leaves unaltered only the symbols 15, 26, 37, 48. In 
the former case, ai, . . . , a4 is a permutation of 1, 2, 3, 4. 
If this permutation is the identity, a permutes only 5, 6, 7, 8, 
and, after applying a transformation not altering P1234, we 
may take <r = (78) or (56) (78), whence S leaves 8 or 4 symbols 
unaltered. In the contrary case, we may assume that a is the 
product of (12) or (12) (34) by a substitution ci on 5, 6, 7, 8. 
If (Ti is the identity, we transform by (18) (27) (36) (45) and are 
led to the preceding case. There remain the cases 

<r = (12)(56), (12)(56)(78), (12) (34) (56) (78), 

for which 5 leaves unaltered 4, 4 or 8 symbols, and the case 
(12) (34) (56), which is equivalent to the second case. 

Theorem 8. There are exactly 4, 8, 16 or 28 real hitangents 
to a real guar tic curve without singular points* 

189. Real Lines on a Cubic Surface. If we adjoin to the 
domain one root of the equation upon which depend the 28 
bitangents to a quartic curve, the group reduces to a subgroup 
simply isomorphic f with the group of the equation upon which 
depend the 27 straight lines on the related cubic surface ( 184). 
The substitutions of period 2 of r which leave one symbol 
fixed leave unaltered 3, 7 or 15 of the remaining 27 symbols. 

Theorem 9. There are exactly 3, 7, 15 or 27 real straight 
lines on a general real cubic surf ace. X 

* It is then not of the tyf)e excluded in the proof of Theorem 1. The group 
discussion by Maillet {I.e., p. 323) is incomplete, as it fails to exclude the case of 
no real bitangents. Weber, Algebra, 2d ed., vol. 2 (1899), devotes pages 458-46.') 
to a proof that no other than these four cases can occur, and three pages to a proof 
that all four cases actually occur. For geometrical treatments see Zeuthen. 
Malhetnaticshe Annalen, vol. 7 (1874), p. 411; Salmon's Higfier Plane Curves, 
p. 220. 

t Note that the quotient of the order 288 -7! of T in 187 by 28 is the order of 
the group T in 183. 

t That these four, but no other, cases actually occur was shown by ScMdjii, 
Quarterly Journal of Mathematics, vol. 2 (1858), p. 117. 



190) ACTUAL GROUP FOR THE BITANGENT8 367 

190. Actual Group G for the 28 Bitangents. We have seen 
that G is a subgroup of the known group V and have applied 
this fact to the study of the reaUty of the bitangents. It is a 
proper conclusion of this investigation to give the proof that G 
is identical with r in case the coefficients of the quartic equa- 
tion are independent variables. For this purpose we need the 
following Lemmas, the proof of which differs from that given 
by Weber * mainly in a slight change pf notation made in the 
interests of symmetry, and in the elaboration of the proof of 
Lemma 2. 

Lemma 1. Given the seven bitangents of any Aronhold set 
of bitangents to a quartic curoe without singular points, we can 
determine rationally the remaining bitangents. 

Let (10) be the given Aronhold set and write Xi for [t8], 
and Xii for [y], i<8,y<8. We are to express the Xi] rationally 
in terms oi x\, . . . , xt. Since xia:23, 0:2X13, xzX\2 are pairs 
18 23, 28 13, 38 12 of a Steiner set of type (9), we may mul- 
tiply X23, etc., by constants as in the derivation of (6), and 
obtain the equation /=0 of the curve in a form such that 

(12) f=^2XlzX2Xi2U^ = ^^X\2X\X2^1^ 

= 'iXiX23X2Xi3'U^, 

where 

(13) U=XiX23-\-X2Xi3-\-X3Xi2, ^^ =^1X23 -X2iCl3+a:3iCl2, 

w = rciX23 +X2X13 a^sici 2. 

Similarly, X2X12 and x^ia are pairs of a Steiner set (8) with 
g = S, h = l, and 

(12') f= ^2X12X4X14- q^f 

where ^ is a quadratic form. Then by (12i), 

^2Xi2{x3Xi3-X4Xu) = ('U-q)iu-\-q). 

If one of the factors on the right were divisible by X2 and the 
other by X12, the intersection of X2 and X12 would be on a = 

-^ Algebra, ed. 2, vol. 2, 1899, p. 442. Down to this point in our eqxwiti<a of 
tiie thepry, we have made only an indirect use of Weber's treatment. 



368 BITANGENTS TO A QUARTIC CURVE (Cu. XIX 

and hence be a singular point of /=0 by (12i). Hence, after 
changing the sign of q if necessary, we may assume that 

u q = 2\\X2X\2y 

where Xi is a constant not zero. Then 

u-\-q = 2{xzXi^-XaXi^/\i. 

Adding and replacing u by its expression (13), we get 

(14) XAX\A=X2X\z+\i^X2Xi2-\\{ XiX2i-\-X2Xiz-\-X-iXi2)' 

In the same manner (or by permuting 1, 2, 3 cyclically), 
we get 

, ,- [XAX2^=XiXi2-\-y^2^XzX2^-y<2{xiX2^X2XiZ-\-XzXi2), 

(14) \ 

\x^Xz^=X2X2Z-\->^7?X\Xi:i\z{XiX2Z-\-X2XizX^l2), 

where X2 and X3 are new constants not zero. 

To determine the X's, divide equations (14') by X2 and X3, 
respectively, and then add. We get the identity 

(15) ^y^+^\ =xY^'+X3:^i3-2:.23U:^23/, 

where / = X2X3+a:2/X3. Since xa, x\, 0:23 occur in the Steiner set 
(8) with g = 8, /f = 3, and no two are paired, they are not con- 
current (proof of Theorem 3) ; similarly, no three of the Xi con- 
cur. Hence /, x^, x\ are concurrent, so that x^ is a linear func- 
tion of / and x\. But 

(16) Xi = a\Xi+a2X2-\-azXz, 

where the a's are known constants each not zero. This sum 
must vanish for 1 = 0, a;i=0, whence a3 = a2X2X3. Thus 

X2 = Ai03, = h\a2, 

A3 

where h\ is a new constant. Then, by (15) and (16), 

h,xJ'^+'^-h,X2^ =^1 N+^-//i(2+aiA0x23l. 
\ X2 X3 / L a2 a3 J 



1901 ACTUAL GROUP FOR THE BITANGENT8 3C9 

Hence, if i^i is a new constant, 

(.17; - \-- niX23 = T-xi, 

A2 A3 Hi 

a2 as 
Permuting 1, 2, 3 cyclically, we get 

k2X4 =12 ^.^3 _ ^2(2 + a2//2)xi3, 

as ai 

hx, = ^-\-'^-h3(2-taMxi2. 
ai a2 

Since the three expressions for x^ must be identical, the three 
^'s are equal and will be designated by k. Also, 

= 1 +aihi{2+aihi) = (1 +a<A,)2 {i = 1, 2, 3), 

so that ki= l/at. Thus 

(18) to=^+2l2+2i?. 

ai a2 as 

Since Xi is derived from Xs by permuting 1, 2, 3 cyclically, we 

have 

a2 as ai 

Al = , A2= , A3 = . 

as ai a2 

Permuting 1, 2, 3 cycUcally in (17), we get 

(19') r^ \---=-kaiXi, 

A2 A3 fli 

X34 , Xi4 :J:13 , a;i4 , X24 X\2 , 

T \r=-. ka2X2, T" + T~ = "~; kasX3. 

A3 Al a2 Al A2 <l3 

Adding and employing (18) and (16), we get 

-^-\-^+X^=-k{aiXi-\-a2X2+a^xz). 

Al A2 A3 

Hence 

(19) ^J^-k[a2X2-k-a3Xs), ^ = ~-k{axXi+aiXi), 
Xi ai ' X2 a2 

*34 X\2 . . 

T- = )i{a\X\-\-aixi). 

Xs ^3 



370 BITANGENT8 TO A QUARTIC CURVE [Ch. XIX 

We may employ xs, Xq or xr in place of X4 in the preceding 
discussion. Instead of (16), we use 

(20) Xi = aiiXi-\-ai2X2+ 043x3 (i = 4, 5, 6, 7), 

where the o^ are known constants. Corresponding to (18) 
are 

(21) ki{aiiXi+at2X2-\-ai3X3) = -{-+ (j = 4, 5,6, 7), 

an 0x2 Ck3 

in which ki, . . . , ki are constants to be determined. If the 
determinant 

J_ i_ i_| 
fli <2 a3 I 

were zero, where for example i = 4, 5, 6, then the corresponding 
Xi would be linearly dependent, whereas they were shown not to 
concur. Hence the ratios of four numbers h, . . . , I7 are de- 
termined by 

(22) k+lL+h+k^o 0- = i,2.3). 

^ ^ a4j a5i aoj ajj \j y , / 

Multiplying the fth equation (21) by /< and summing, we 
see that the new right member is identically zero, so that 

7 7 7 

(23) ^kika,i = Q, 2^,/ifl<2 = 0, ^kUii = ^. 

<-4 f-4 i4 

These equations determine the ratios of the ^'s, one of which 
is arbitrary and may be taken to be unity. Then (21) deter- 
mine X23, x\3, x\2, while (19) and the corresponding formulas 
determine xu, X2i, xst (j = 4, . . . , 7) rationally. We now 
have all the bitangents except 45, 46, 47, 56, 57, 67. But if 
we had used X5, xa, xi in place of x\, X2, x^, we would have ob- 
tained rationally all except 12, 13, 14, 23, 34, 34. The two 
steps together give all the bitangents a:^. 

Lemma 2. If we choose seven straight lines in a plane in a 
sufficiently general manner, we can determine rationally the equa- 
tion of a quartic curve without singular points for which the seven 
chosen lines form an Aronhold set of bitangents. 



190) ACTUAL GROUP FOR THE BITANGENT8 371 

For proof, we have only to reverse the argument made 
for Lemma 1, now taking the a^j as independent variables. 
Then 0:23, X13, a:i2 are determined rationally in terms of the o^ 
by means of (21)-(23), and then X4, X5, xe, xj, xu, xst, x^i, xjt 
are determined rationally by means of (20), (19) and the analo- 
gous equations mentioned above. 

Substituting the expressions for X23, ^13, X12 into (12i; 
and (13), we obtain the equation /=0 of a quartic curve, whose 
coefficients are rational functions of the Gfj. Its discriminant 
is not identically zero, since we saw in the proof of Lemma 1 
that we can deduce equations (21)-(23) from the equation of a 
quartic curve without singular points. 

For i = 4, (20) and (21) give (16) and (18). From these and 
(19) we get (19'). Substitute the left member of the latter 
into (15). In the resulting term 0:4^23 Ai, replace X4 by its 
value (16); in the term kaiXiX4, Tep\a.ce kxA by its value 
(18). We obtain the right member of (15). Hence we have 
(15) and the equations derived from it by permuting 1, 2, 3 
cyclically. From the relation between (14') and (15), it follows 
at once that (14) and (14') hold. Define q by the equation 
preceding (14). Then uq has the value indicated, so that 
(12') follows from the equation written below it. The equa- 
tions obtained from (12') by replacing 4 by 5, 6, 7 follow 
similarly from (20), (21) and the equations of type (19). Hence 
xa, X5, xg, X7 form with 'x\, X2, X3 an Aronhold set. 

Theorem 10. If the ratios of the 15 coefficients of a ternary 
quartic form f are independent variables and if R is the domain 
of the rational functions of these 14 ratios with rational coefficients, 
the group G for R of the equation E{^)=0 upon which depends 
the determination of the 28 bitangents to f=0 is the group V 
q/"187. 

It was proved in 187 that every substitution of G is in f. 
To prove the converse, it is sufficient, in view of the Corollar>' 
in 149, to show that every rational relation with coefficients 
in R between the roots of E{^)=0 is pre.served by each sub- 
stitution of r. To this end, let {1, . . , {7 be the roots corre- 
sponding to the bitangents 18, ... , 78. Since the latter 



372 BITANGENTS TO A QUARTIC CURVE [Ch. XIX 

form an Aronhold set, the remaining 21 roots ^ = $^ {i,j=l, 
. . . , 7; iy^j) are rational functions of fi, . . . , {7 with 
coefficients in R (Lemma 1). If ci, . . . , cu are the ratios 
of the coefficients of /, any rational relation between the roots 
with coefficients in i? is of the type 

*Ul, . . , ?7, ?12, . . . , $67, Cl, . . . , Cl4)=0, 

where <l> is a rational function of its arguments with rational 
coefficients. First, we replace the f</ by their rational expres- 
sions in terms of the $*, Ct. Next, we replace each Ck by its 
rational expression (Lemma 2) in terms of the coefficients $, 
Tfii (i = l, . . . , 7) oi the seven bitangents of our Aronhold 
set. But these 14 quantities can be chosen at will. Hence 
after our replacements, relation $ = becomes an identity 
in the $, w. Thus <l> = remains true if we substitute for 
$1, . . . , $7 the seven roots in any order of any Aronhold set, 
provided of course we replace each $y by the root which arises 
from it by our substitution. But r is the group of all such sub- 
stitutions. Hence G = T. 

Corollary. The adjunction of one root of a certain 
equation of degree 36 reduces the group of the equation for the 
28 bitangents to the group of the general equation of degree 8. 

In fact, the subgroup ( 187) of r is simply isomorphic 
with the symmetric group on 8 letters and is of index 36 under r. 

Since we know the generators of r and the representation of 
each substitution of r in terms of the generators (end of 187), 
we can prove by a straightforward argument that F is a simple 
group (Weber, I.e., pp. 454-6). 

191. Symmetrical Notation for the Bitangents to a Quartic 
Curve. The separation of the Steiner sets into two types and 
likewise for the Aronhold sets was due to the lack of symmetry 
in the notation of Hesse and Cayley and not to a geometrical 
difference. A perfectly symmetrical notation was discovered * 

Riemann, Werke, 1876, p. 471. Weber, Theorie der Abelschen Functionen 
vom Geschlecht 3, Berlin, 1876, p. 82. Clebsch, " Ueber die Anwendung der 
Abelschen Functionen in der Geometrie," Crelle, vol. 63 (1864), p. 211, who 
used the notation (xi, xj, Xj; yi, yt, yi). Appell and Goursat, Thiorie des Fonclions 
Algibriques, 1895, p. 511. 



191] SYMMETRICAL NOTATION FOR THE BITANGENT8 373 

in connection with the theory of theta functions of odd character- 
istics 

(24) (xi^'i X2>'2 xzyz), 
where each rcj and >>< is or 1, and 

(25) x\y\-\-X2y2-\-xzy:i = \ (mod 2). 

11x^ = 1 (mod 2), the congruence determines yz in terms 
of x\, y\, X2, y2, so that there are 2* such sets of solutions. If 
a;3=0 (mod 2), 

(26) xiyi+X2y2 = l (mod 2) 

has the four sets of solutions 0:2 = 1, y2 = lxiyi, and the two 
sets 0:2=0, y2=0 or 1, xi^yi = l; since >'3 = or 1, we obtain 
2X6 sets of solutions of (25) with aca^O. Hence there are 28 
symbols (24). 

Theorem 11. The 28 bitangents to a general quartic curve 
can he designated by the 28 symbols (24) in such a way that the 
8 points of contact of the four bitangents 

A = {aibi 02^2 ^363), B = {cidi c^2 csds), 

C = (xiyi a;2>'2 xsyz), D = {ziwi Z2IV2 Z3W3) 

are on a conic if and only if 

(27) o,+ci+x,-f2=0, bt-\-di-\-yt+Wt^0{mod2) (t = l,2,3). 

This theorem, which leads to a symmetrical notation for the 
bitangents and presents the problem of the bitangents in a form 
suitable for extensive generalizations ( 192), was deduced in 
the papers last cited from the theory of abelian functions. 
We shall here give a very elementary proof,* depending upon 
two lemmas. 

Lemma 3. If A and B are any two distinct symbols (24), 
there exist exactly five pairs of symbols Co, Do ; . . . ; C4, D4, 
distinct from each other and from A and B, such that the sums of 
corresponding elements of the symbols A , B, Cj, Dj are all even, 
as in Theorem 1 1 . 

* Dickson, Bull. Amer. Math. Soc., vol. 20 (1914), pp. 403-4. 



374 BITANGENTS TO A QUARTIC CURVE [Ch. XIX 

This lemma implies that the sums of corresponding elements 
of Co, Do, Cj, Dj, are all even. Thus the set of six pairs deter- 
mined by A and B is identical with the set of six pairs deter- 
mined by Co, Do. For such sets of six pairs, properties (A) 
and (B) of 186 therefore hold. The fact that also property 
(C) holds and hence also Theorem 11, may be stated as 

Lemma 4. The sets AB, CD, . . . and AC, BD, . . . 
have no further symbol in common. 

In the proof of these two lemmas, it suffices to consider 
symbols A, B having h^^ds (mod 2), and hence (after inter- 
changing A and B if necessary) with 63 = 0, d3 = l. For, if 
hz=dz, but hiT^di, the symbols A' = {azhz ^2^2 dibx) and B' , 
derived from A and B by interchanging the first and third pairs 
of elements, lead by the proof below to just five pairs C'j, D'j, 
from which we derive the required Cj, Dj, by interchanging 
the first and third pairs of elements. Next, if each bi=di, 
then ai^ci, for example, and we proceed as before with 
A* = {biai ^2^2 &3<i3), B* = {dici ^2^2 ^3^3). 

To prove Lemma 3, we may therefore assume that 63 = 0, 
d3 = l. If C and D are symbols for which congruences (27) 
hold, then y-i-\-W3 -{-1=0 (mod 2), so that either y3=0 or 
W3=0. Since the mutual order of C and D is immaterial, we 
may set 3^3 =0, whence 103 = 1 (mod 2). The conditions that 
C and D shall satisfy the condition (25) for a symbol are (26) 
and ziWi-\-Z2'W2-hz3 = l. By (27), the latter becomes 
2 

(28) ^{ai-\-Ct-\-Xt){bt-\-dti-yt)-\-a3+C3-\-X3 = l (mod 2), 

which determines X3 in terms of xi, yi, X2, y2. There are six 
sets of values of the latter which satisfy (26). One of these sets 
is Xi=ai, yi=hi {i \i 2), whence X3=a3, C=A (and hence 
D=B), since 

(29) C3 = l+Ciii+C2<f2 (mod2). 

Since this set is to be excluded, Lemma 3 is proved. For use 
in the proof of Lemma 4, we shall exhibit the five pairs Cj, Dj. 
In view of aibi-\-a2b2 = lx by and 62 are not both even. 



192) FURTHER PROBLEMS OF CONTACTS OF CURVES 375 

After interchanging the first and second pairs of elements 
in all of our symbols, if necessary, we may set 62 = 1. Then 
we may set 

A=(a h ah-\-\ 1 e 0), B = {cidi 7^2 C3I), 

in which, as well as below, cz is given by (29). The sets of solu- 
tions of (26), other than the above excluded set, are evidently 
the five sets of the first four elements in Co, ... , C4 below. 
After determining xz for each by use of (28), we see that the 
five pairs of symbols specified in Lemma 3 are 

C*=(l I kOzi, 0), Z>=(a+c,+l, 6+di+l, oA+l+c+A, dt+X, c+c+s*, 1) 

Cj=(a+1, 6, ai+6+1, 1, c+rf,+W,, 0) A=(ci+1, rfi, ct-\-b, d,, c,+rf,+W,, 1); 

Ci={a, b+l, ab+a+l, 1, e+Ci+adt, 0), Z),= (ci, di+l, Ct+a, dt. c+Ci+ad,, 1); 

C= (a+l, b+l, ab+a+b, 1, e+a, 0), Dt= {ci+1, di+1, Ci+a+b+l, dt, c,+a, 1) 

[a= Ci+di+d,+ l+adt+bd,]. 

To prove Lemma 4, we have to show that if C is one of 
these 10 symbols and E is one of the 8 not paired with C and 
not identical with C, the new set AC, BD, . . . does not con- 
tain E. If it did, there would be a symbol paired with E whose 
elements are the sums of corresponding elements of i4, C, E. 
But we readily verify that condition (25) is never satisfied for 
this symbol paired with E. After treating the cases in which 
C = Ci, we need not consider the cases C = Dt, since if P and Q 
form any pair of our ten symbols, A-{-B = P-\-Q = Ct+Dt 
imply A-\-Di-\-P=A+Ci-\-Q. Hence, treating C(ife = 0, 1) 
together, we need consider only six cases with = 2, four 
with C = C3, two with CC4, and C = Co, E=Di. For example, 
^-f C2+G = (0 1 6-fi^ 5 0) is not a symbol satisfying (25). 

192. Further Problems of Contacts of Curves. The pre- 
ceding symmetrical notation for the bitangents to a quartic 
curve is in accord with that used by Steiner * and Clebsch f 
in their treatment of a series of problems on contacts of curves. 

*JpurnaIfUr Math., vol. 49 (1855). 
t Ibid., vol. 63 (1864). 



376 BITANGENTS TO A QUARTIC CURVE (Ch. XIX 

Let Cm be a real plane curve of order n having no double point. 
Set 

/> = i(n-l)(-2), Rj, = 2^''-^-2'-\ 

There are Rp curves of order w 3 having simple contact with 
C at (n 3)/2 points. The determination of these curves 
depends upon an algebraic equation E of degree Rp whose roots 
are designated by {xiyi xiji . . . Xpj^, where X\, . . . , yp 
form any set of integral solutions of 

x\y\-\-X2y2-\- . . . \-Xpyp=\ (mod 2). 

The simplest case is = 4; then /> = 3 and the problem is 
that of the i?3 = 28 bitangents to a quartic curve. For w^4, 
Clebsch proved that, if \i is any positive integer ^ Rp for which 
m( 3)/2 is an integer, the points of contact of Cn with the 
/i curves corresponding to the roots 

{x\y\ . . . x'py'^), . . . , (xi(''>>'i<''> . . . Xp^'^^yp^^) 
lie on a curve of order /i( 3)/2 if the congruences 
x\^-x'\^- . . . +a;p<^>=0, /p+3'"p+ . . . +>'p^^=0 (mod 2) 

(p = l, ...,/>) 

hold simultaneously. For = 4, the first case /i = 2 is evidently 
trivial, while the next case /x = 4 is the one treated in Theorem 11. 
The group * of equation E can be represented as a subgroup 
of the abelian f linear homogeneous group on 2p variables 
with integral coefficients taken modulo 2. Its substitutions of 
period 2 are conjugate to certain simple types, from which 
fact in connection with the Corollary in 188 we find that the 
number of real roots of equation E is one of the numbers 

22p--(^ = l, ...,/>), 2=^-2^-i-2'-^ (y = 0, 1, . . . , tt), 

where Tr = p/2 or {p \)/2, according as p is even or odd. For 
= 4, we again get the number of real bitangents to a quartic 
curve ( 188). For = 5, we have /> = 6 and see that, of the 
2016 conies tangent at 5 points to a quintic curve without 

* Dickson, Annals of Math., ser. 2, vol. 6 (1905), p. 146. 
t Leaving invariant a certain bilinear function of two sets of cogredient vari- 
ables. It is not a commutative group. 



192] FURTHER PROBLEMS OF CONTACTS OF CURVES 377 

double points, either all, 1024, 512, 480, 256, 128, 96, 64, 32 
or none are real. 

These results, obtained so easily by group theory, are in 
complete agreement with those obtained by an elaborate geo- 
metrical proof by Klein.* 

The case in which the curve C has w(w 3)/2 double points 
was treated by Clebsch f and from the group standpoint by 
Jordan. f The latter treated also the group of the equation 
upon which depend the 16 singular tangent planes to Rum- 
mer's quartic surface with 16 singular points {I.e., pp. 313-5); 
also several technical problems of contacts of curves and the 
problem of the 16 straight lines on a quartic surface with a double 
conic (I.e., pp. 305-313). 

The various geometrical problems treated or mentioned in 
this Chapter and the preceding one have led to linear con- 
gruence groups. Such groups enter into the majority of the 
questions treated in Jordan's Traite des Substitutions and form 
the exclusive subject of Dickson's Linear Groups. 

* Math. Annalen, vol. 42 (1893), p. 3, p. 26; Rietnann'sche FUkhen, II (1892), 
pp. 117-255. 

t Journal fUr Math., vol. 64 (1865). 

X Traits des Substitutions, 1870, pp. 331-3. 



CHAPTER XX 
MONODROMIE GROUP 

193. Definition of the Monodromie Group M. Consider 
an algebraic equation F{z, k)=Q in z whose coefficients are 
rational functions of the complex variable k. Let zi, . . . , Zn 
be the roots of F{z, ko) =0, where ^o is a constant. Let k vary- 
continuously from this initial value in any manner, but finally 
return to the same value ^o (ie., let the point representing k 
in the complex plane describe any closed path starting from and 
ending with the point representing ^o). Then the roots vary 
continuously and, after the circuit, take on their initial values 
in the same or a new order. Thus to each closed path corre- 
sponds a substitution on the roots. 

For example, if k describes a circle around the origin, the 
roots of s^ = 2k are interchanged. 

Two circuits may be combined into a single third circuit 
to which corresponds the product of the two substitutions 
corresponding to the two circuits. Hence the substitutions 
corresponding to all possible circuits form a group M, called 
the monodromie group * of F{z, k)=0 with respect to k. It 
was first studied by Hermite and Jordan. f 

194. Monodromie Group an Invariant Subgroup of the 
Galois Group. Let <^ be a rational function of k and the roots 
zi, . . . , Zn, and let <}>, </>', <{>", ... be the functions derived 
from <f> by the various substitutions of If . If ^ = 0' = </>" = . . . ^ 
4> is said to possess monodromie with respect to k. This is 
evidently the case with any <^ which equals a rational function 

* " Group in the function-theoretic sense," by Klein-Fricke. EUiptischen 
Modulfundionen, vol. 1, IS(K), p. 132; applications in vol, 2, 1892, p. 53, p. 599. 
t Traile des StibstUulions, 1870, pp. 277-9. 

378 



1941 MONODROMIE GROUP 379 

of k, whether or not the coefficients of <t>(k, Zi, ..., 2) in- 
volve irrational constants. 

Let R be the domain defined by k and the coefficients of 
the powers of z in F(2, k). Any rational function <f> of the 
roots with coefficients in R, which equals a quantity in R, 
equals a rational function of k and hence is unaltered by every 
substitution of M. Then, by the Corollary in 149, Af is a 
subgroup of the Galois group G for R of the equation F = 0. 

Moreover, M is an invariant subgroup of G. For, let <t> be 
a rational function of the roots with coefficients in R which 
belongs to the subgroup M of index p under G. Then is a 
root of an equation E of degree v with coefficients in /?, so that 
<t> is an algebraic function of k. But <^ p>ossesses monodromie 
with respect to k. Hence <^ is a rational function f{k) of k 
with perhaps irrational coefficients. Replace the coefficients 
of f{k) by independent variables and substitute the result- 
ing expression in place of </> in E, and let the result be an 
identity in k. We obtain certain algebraic equations which the 
variable coefficients of f{k) must satisfy. Adjoin to R all of 
the roots of these numerical equations. Since <t> is in the en- 
larged domain, G reduces to a subgroup of M, necessarily M 
itself, since the group of monodromie is evidently unaltered 
by the adjunction of constants. But the adjunction of all of 
the roots of a second equation reduces the Galois group of the 
first equation to an invariant subgroup ( 167). A number of 
such adjunctions reduced G to Af ; whence M is invariant in G. 

For example, the Galois group for Rik) of 2* 2yt*=0 is the symmetric 
group Gt, since the equation is irreducible in R and the product of the dif- 
ferences of its roots is QVZk*. The only circuits causing a permutation 
of the roots are those aroimd the origin. Hence M is the cyclic group of 
order 3 and is invariant in G. 

195. Applications of Monodromie. Jordan * employed 
monodrorriie to determine the Galois group of the equation for 
the n-section of the periods of elliptic f and hyj>erelliptic 
functions with 2/> periods. For the case of the trisection of 

Train des Substitutions, pp. 337-369. 

t Cf. K. Weber, Elliptische Functionen, 1891, p. 219. 



380 MONODROMIE GROUP [Ch. XX 

the periods of hyperelliptic functions of four periods, the group 
is the quaternary abelian linear group modulo 3 and is iso- 
morphic * with the group for the equation of the 27 lines on a 
cubic surface (183). 

Monodromie has been applied to linear differential equations 

^T.+/.(|S+ +/.(*) -0. 

For simplicity, the coefficients /(^) will be assumed to be rational 
functions of the complex variable k. Let 2i, . . . , 2 be a set of 
linearly independent solutions (integrals) and let ^o be a constant 
such that each 2 is an ordinary power series in ^ ^o- If now the 
point representing k describes a closed path starting from and 
ending with the point representing ^o, as in 193, the set of 
solutions zi, . . . , Zn becomes a set of solutions z'l, . . . , z', 
which are therefore linear functions of zi, . . . , Za with con- 
stant coefficients: 

z'l=ailZi+ . . . +ln3n, . , z'n=an\Z\-\- . . . +ann2n. 

With the chosen circuit is thus associated a linear transformation 
(75). The transformations obtained from all such circuits 
form a Unear group, called the monodromie group M of the 
differential equation. 

This group M is finite in case the integrals zi, . . . , z are 
algebraic functions of k. The theory of finite linear groups 
(Part II) is therefore applicable to the problem f of the deter- 
mination of all linear differential equations whose integrals are 
all algebraic. In the case just mentioned, M coincides with 
another important group G, which we proceed to define; but, 
in general, M is only a subgroup of G. 

According to Picard and Vessiot, the transformation group 
G of a linear differential equation with the linearly independent 
solutions zi, . . . , Zn possesses the following two characteristic 
properties (analogous to properties A and B of the Galois group 
of an algebraic equation, 149) : 

If a rational function F of Zi, . . . , z and their derivatives 

* Jordan, I.e., p. 369; Dickson, Linear Groups, pp. 306-7. 
t Jordan, Jour, fur Math., vol. 84 (1877), pp. 89-215. 



1961 QUINTIC EQUATIONS, FORM PROBLEM 381 

remains unaltered (as a function of k) by all the transformations 
of G, then F equals a rational function of k. 

Conversely, if such a function F equals a rational function 
of k, it remains unaltered by all the transformations of G. 

For the development and application of these concepts, see 
C. Jordan, Cours d' Analyse, vol. 3, 1896, pp. 193, 203, and 
L. Schlesinger, Handhuch der Linearen Differentialgleichungen, 
vol. 2, I, 1897, pp. 1-226, especially pp. 71, 96-102; vol. 2, II, 
1898, pp. 148-159. 

196. Quintic Equations, Form Problem. In the " form 
problem " for the icosahedral group, we are given the values 
of the fundamental invariants T, H,/, consistent with the relation 
between them ( 105, E), and require the values of the variables 
xi, X2. However, we desire primarily only their ratio z=xi/x2. 
Hence, given Z, we seek the values of z for which 

This icosahedral equation of degree 6o is remarkable both 
on account of the property that all its roots are linear fractional 
functions of a single root and the fact that the roots of any 
quintic equation can be expressed in terms of radicals and a 
root 2 of (1). In his Vorlesungen iiber das Ikosaeder, 1884, 
KJein therefore regards z as a new fundamental irrationality, 
a stage higher in algebraic complexity than radicals. More- 
over, z can be expressed in terms of elliptic modular functions 
{ibid., p. 132). 

Naturally there are many resolvent equations of degree < 60. 
For example, if we use the tetrahedral invariant t oi 105 and 
set r = i^/f, we get the resolvent of the fifth degree 

r(r2-l0r+45)2 + 1728(Z-i)=0. 

The form problem of the tenary linear group of order 168 ( 1 23, 
J, 125) is connected in a similar marmer with the equations of 
degree 7 whose Galois group is the simple group of order 168. 

For references to these and related subjects, consult Encyklo- 
padie der M athematischen Wissenschaften, vol. 1, I, pp. 533-552, 
513-4; Weber, Algebra, ed. 2, vol. 2, 470-496, 530-550. 



SUBJECT INDEX 



(The numbers refer to pages) 



Abelian group defined, 62, 87 
Abstract definitions of groups, 143 

group defined, 52 
Alternating group, 18, 43, 166 
Anharmonic group, 65 
Arithmetic substitutions, 12 
Automorphisms of a group, 46 
Average number of letters in all the 

substitutions of a transitive group, 32, 
of an intransitive group, 73 
Axial group, 65 

Base of an abelian group, 89 
Bauer's theorem, 124 
Bertrand's problem, 166 

Cayley's table, 64 

theorem, 64 
Central of a group, 68 

quotient group, 68 
Characteristic subgroups and character- 
istic operators, 71, 109 

Class of a substitution and of a sub- 
stitution group, 47 
Cogredient isomorphisms, 68, 76 
Commutative operators and commuta- 
tive groups, 54, 62 

substitutions, 2, 18 
Commutator quotient group, 69 
Commutators and commutator sub- 
group, 68 

Complementary groups, 67 
Complete group, 46 

set of conjugates, 22 



Composite groups, 43 

Conformal groups, 107 

Congruence groups, 10 

Congruent elements, 67 

Conjoints, 37 

Conjugate operators or subgroups, 59 

substitutions and substitution 
groups, 5, 21 

Construction of groups of order ^, 138 

with invariant subgroups, 59 

Contragredient isomorphisms, 76 
Co-set and augmented co-set, 24, 66 
Cross-cut of two groups, 23 
Cyclic group, 7, 54 

substitution, 16 

Degree and order of a group, 2 

of a substitution, 8, 16 
Derived groupw, 69 
DicycUc groups, 62, 170 
Dihedral groups, 61, 168 
Direct product, 13, 77 
Divisible groups, 117 
Divisor of a group, 3 
Double co-set, 25 

holomorph, 46 

Doubly transitive groups, 40, 164 

EUementary divisors of the order of an 

abelian group, 88 
Elements of a group, 52, 68 
Equivalent or congruent dements, 07 
Even or positive subsUtutkms, 17 
Examples of groups, 1 



383 



384 



SUBJECT INDEX 



Factor groups, 176 
Factors of composition, 174, 184 
Four-group, 65 
Frobenius's theorem, 77 
Fundamental characteristic subgroup, 
110 

Generalizations of the groups of the 

regular polyhedrons, 152 
Generating substitutions of a group, 7 
Group defined, 2, 52 

generated by two operators having 
a common square, 143 

of a function, 5 

isomorphisms, 46, 95, 101, 134, 

160 
the square, 4 

property, 86 

Groups involving no more than four 

letters, 41 
only abelian subgroups, 112 

of degree five, 45 

movements of plane figures, 9 

subtraction and division, 15, 81 

the regular polyhedrons, 147 

represented by matrices, 13 

whose orders are px)wers of prime 
numbers, 118 

divisible by 2 hut not by 4, 

66 

having simple abstract definitions, 
143 

Hamilton groups, 115 
Holomorph of a group, 40 

Icosahedron group, 150, 158 
Icosian calculus, 159 
Identity, 2, 53 

Imprimitive substitution groups, 3S 
Independent cyclic subgroups, SS 
Index of a subgroup, 23 
Indivisible group, 117 
Inner isomori)hi3ms, 76, 183 
Insolvable grou.js, 174, 180 
Intransitive substitution group, 31 
Intrinsic and relative properties of 
operators, 159 



Invariant abelian subgroups of a prime- 
power group, 120 

operators in the group of isomor- 
phisms of an abelian group, 101 

subgroups, 21, 66 
Invariants of an abelian group, 88 
Inverse of a substitution, 11 

an element or operator, 54 

Isomorphisms defined, 33, 160 

of the alternating and the sym- 
metric groups, 166 

an abeUan group, 101 

Jordan's theorem, 35 

Kuhn's theorem, 37 

Lagrange's theorem, 23 
Left co-sets, 66 

Maximal subgroup, 39 

Metabelian group, 68 

Metacyclic group, 12 

Movements of the regular polygon, 9 

Multiple isomorphisms, 34 

Multiplication of substitutions, 2 

Multiply transitive groups, 40 

Nebengruppen, 24 

Negative or odd substitutions, 17 

Number of elements of a given order in 

an abelian group, 93 
operators in a set of independent 

generators of a group of order />"* 

is invariant, 127 
subgroups in a prime-power 

group, 123 
of order p" in any group, 125 

Octahedron group, 149, 154 
Octic group, 4 

Operator or operation of a group, 68 
Order of a group and of an operator, 
2,53 

a substitution^ 8 

the product of n operators, 70 

Outer isomorphisms, 76 



SUBJECT INDEX 



385 



Perfect group, 69 

Period of an element of a proup, 53 
Permutations and substitutions^ 36 
0-subgroup, 71 
P-isomorphisms, 134 
Positive and negative substitutions, 16 
Power of a group, 32 
Primary groups, 1 18 
Prime-power groups, 118 
Primitive and imprimitive substitu- 
tion groups, 38 
Product of two substitutions, 2 

Quadratic group, 65 
Quaternion group, 62 
Quotient group, 34, 66 

Rank of an abclian group, 92 
Relative and intrinsic properties of 

operators, 159 
Regular substitution group, 35 
Representation of a group as a regular 

substitution group, 63 
an abstract group as a transitive 

substitution group, 81 
Right co-sets, 66 
Roots of operators of an abelian group, 

114 

Self-conjugate subgroup, 21 
Series of composition, 177 
Set of independent generators of a 
group, 9, 90, 127 



Similar substitutions and similar groups, 

21 
Simple group, 43 

isomorphisms and simply isomorphic 
groups, 33, 73, 95 

Simplicity of the alternating group, 43 

Solvable group ,174 

Subgroup, 3 

Subgroups and quotient groups of an 

abelian group, 99 
Substitution and substitution group, 

1,2 

groups of degree five, 45 
Substitutions commutative with a 

given substitution, 19 
Sylow's theorem, 27 
Sylow subgroups, 27, 181 
Symmetric group, 1, 3, 166 
Systems of imprimitivity, 38 

Tetrahedral group, 147, 152 

Totient of a number, 11 

Totitives of a number, 95 

Transform of a substitution and of a 
substitution group, 20, 57 

Transitive constituent of an intransi- 
tive group, 33 

substitution group, 31 
Transitivity of the symmetric group, 40 
Transposition, 16 

Vierergruppe, 65 



PAKT n 



Abelian groups, canonical form of, 213 " 
Algebraic integer, 241 

Canonical form, 196; theorems on, 
212, 213 

Change of variables, 203 

Characteristic and characteristic equa- 
tion, 205 

CollineatioDS and colUneation-groups, 
198 

Conjugate-imaginary groups, 209 



Determinant of a linear transforma- 
tion, 194; theorems on the deter- 
minants of the transforroatiuns be- 
longing to a finite group, 196, 200, 
202 

Dihedral group, 220, 225 

Diopbantine equation, 227 

Exjuivalcnt groups, 202 
Group-matrix, theorem oo, 268 



386 



SUBJECT INDEX 



Groups, binary, 215 

, linear, 198 

, linear fractional, 201 

of coUineations, 198 

linear transformations of deter- 

ninant unity, 200 

the regular polyhedra, 220 

order /<*, 231 

order /-V, 272 

degree n and class n 1, 274 

, ternary, 235 

Hermitian form, 207 

invariant, 209 
Hessian group, 239 
Homology, 248 

Icosahedral group, 223, 226 

Identical transformation or the iden- 
tity, 196 

Imprimitive groups, 228; theorem on, 
229 

Imprimitivity, sets of, 228 

Intransitive groups, 206 

Intransitivity, sets of, 206 

Invariants (absolute, relative), 258; 
theorems on the number of, 258, 259 

of the binary groups, 224 
ternary groups, 253 

Inverse of a linear transformation, 194, 

210 
Irreducible groups, 211 

Linear transformation, 194 



Matrix of a linear transformation, 194; 
theorems on the matrices of the 
transformations belonging to a finite 
transitive group, 271 

Monomial groups, 229 

Multipliers of a linear transforma- 
tion, 196; theorems on, 197, 257 

Non-equivalent groups, 262; theorem 
on, 271 

Octahedral group, 222, 225 

Order of a linear transformation, 196 

primitive group, 256 

Power of a linear transformation, 196 
Primitive groups, 228; order of, 256 
Product of linear transformations, 195 

Reducible groups, 210 

Regular substitution groups, theorems 

on, 264, 265, 269, 272 
Roots of unity, 239 

Similarity-transformations, 196; the- 
orems on, 197, 202, 233, 234 
Sylow groups, 231 

Tetrahedral group, 222, 225 
Transitive groups, 207; theorems on, 
260, 261 

Unitary form, 210 



PAKT in 



Abel's theorem, 320 
Abelian functions, 373 
linear group, 376-7, 380 
Adjunction, 299, 313-4, 317-20, 323-4, 

a38-41, 379 
Aronhold set, 361, 367-71 
Asyzygetic, 356, 361-3 

Belongs to group, 296-7, 299 
Binomial equation, 281, 312, 316 
Bitangents, 350-76 



Cardan's formulas, 301 
Congruence group, 335, 377 
Conies, 3.53-6, 373, 376 
Conjugate, 297, 305 
Constructions by ruler and compasses, 

321-6 
Contact, 353-77 
Cross-ratio, 283 
Cubic curve, 330-42 

equation, 296, 301-3, 320-3 

surface, 343-53, 366, 380 



SUBJECT INDEX 



387 



Ciuve of order n, 376-7 

Cyclic group, 306-9, 311, 316, 319,326 

Cydotomic equation, 308-10 

DifiFerential equation, 380 
Discriminant, 290, 292, 302, 312 
Domain, 280, 293, 299 
Double point, 329 

six, 345 
Duplication of cube, 321-3 

Elliptic function, 379, 381 

Equality, 294 

Equation with variable coefficients, 292 

of the fifth and seventh degree, 381 
Euler's theorem, 328 

Factors of composition, 306, 311-2, 

317,350 
Fan, 361 
Form problem, 381 

Galois' criterion for solvability, 315 

generalization of Lagrange's theorem, 
298 

theorem, 319 
Galoisian equation, 320 

resolvents, 284, 288 
Gauss' lemma, 309 

General equation, 292, 294, 317, 320, 

372 
Geometrical constructions, 321-6 

questions, 321-77 

Group of an algebraic equation, 286-9, 
294, 300, 3a5, 333-41, 347, 362, 365, 
367-71, 376-81 

a differential equation, 380 

on four letters, 290 

Hessian, 329, 344, 354 

curve, 330-3 

Homogeneous coordinates, 327 
Hyperboloid, 346 
Hyperelliptic functions, 379 

Icosahedral equation, 381 

group, 381 
Imaginary roots, 292, 365 



Inflexion, 330-42 
Integral root, 282 
Invariant, 330, 333, 381 

subgroup, 305-6, 318-0, 378 
Irreducible, 280-1, 289, 292, 294, 297, 

310, 320, 325 

case for cubic equations, 320 
Isomorphic, 304 

Jordan's theorem, 317 

Lagrange's formulas, 307 

theorem, 298 

Linear group, 335-8, 380-1 

transformation, 328, 336, 380 
Lines on a cubic surface, 343, 366 

Monodromie, 378-81 

Notation of Hesse and Cayley, 357 

Primitive root, 308, 325 

Quartic curve, 350-76 

equation, 290, 296, 312-4, 337 

surface, 377 
Quintic equation, 381 

curve, 376 

Quotient group, 306, 318, 337 

Radicals (see Solvable) 
Rational function, 284, 289, 294, 296, 
298, 378, 380-1 

root, 282 

Reality, 341, 365-6, 376-7 
Reciprocal equation, 290-2 
Reducible, 280-1, 289, 291, 308 
Regular group, 306, 309, 316 

polygon, 321-6 

Resolvent, 284, 290, 309-. 312. 350, 

381 
Root, definition of, 292 

of unity, 307-11, 323-6 

Series of composition, 306, 317 
Simple group, 3(Nl, :i.y), 372. 381 
Singular point, H29 



388 



SUBJECT INDEX 



Solvable equation, 301-3, 307, 310-4, 
315, 317, 320, 337, 341 

group, 279, 306, 310, 337 
Steiner sets of bitangents, 354-77 
Symmetric group, 295-6, 299, 305, 315, 

317, 337-8, 364, 372 
Symmetrical form for the equation to 
a quartic curve, 355 

notation for bitangents, 372 
Syzygetic, 356, 361-3 



Tangent cone, 351-3 
Theta functions, 373 
Transitive, 289, 306 
Triangle of reference, 328 
on a cubic surface, 344-6 
Trisection of an angle, 321-3, 326 

Unaltered, 287, 289, 294, 296, 298, 38i 

Values of functions, 282, 314 



AUTHOR INDEX 



(The numbers refer to pages) 



PAST I 



Abbati, P., 84 

Abel, N H., 85 

Alasia, C, 86 

Battaglini, G., 86 

Bertrand, J., 166 

Betti, E., 15, 20 

Bianchj, L., 86 

BAcher, M., 13, 65, 86 

Bolza, O., 33, 65 

Burnside, W., 46, 65, 86, 123, 127 

Capelli, A., 65, 160 

Carmichael, R. D., 28 

Cauchy, A. L., 12, 17, 23, 25, 27, 43, 

47,85 

Cayley. A., 64, 85. 159 
Cebyggv, P. L., 166 
Chapman, H. VV., 84 
Cole, F. N., 86 
Dedekind, R,, 69, 97, 115 
Dickson, L. E., 117 
Dyck, W., 81 
Easton, B. S., 86 
Elder, L., 84, 87 
File, W. B., 68 
Frattini, G., 35, 71 
Frobenius, G., 25, 32, 71, 77, 87, 88, 

113, 118 
Galois, E., 2, 12, 24, 27, 45, 6ft, 85 
Gauss, C. F., 11.8.5,87,89 
Haffiilton, W. R., 45, 62, 115, 147, 150; 

159 



Heffter, L., 93 
Hermite, C, 15 
Hilton, H., 124 
Holder, O., 34, 160, 176, 191 
Huntington, E. V., 85 
Jordan, C, 20, 34, 35, 50, 85,86, 160, 176 
Klein, F., 15 
Kronecker, L., 85, 89 
Kuhn, H. W., 31 
Lagrange, J. L., 4, 23, 84 
Loewy, A., 183 
Lucas, E., 98 
Manmn^, W. A., 50 
Mathieu, E., 106 
Miller, G. A., 65. 114.117,138 
Moore, E. H., 105, 160, 161 
Moreno, H. C, 114 
Netto, E., 86, 93 
Pierpont, J., 65 
Poincar6, H., 84 
Ranum, A., 101 
Remak, R., 176 
Ruffini, P., 84 
Schering, E., 89 
S<5guier, J. de, 2 
Scrrct, J. A., 166 
Stickelbergcr, L.. 87, 88, 118 
Sylow, L., 27, 119 
Sylvester, J. J., 1 1 
Vandermonde, A. J., 17, 84 
Wcbcr, H., 12, 24, 34, 95, 08. 174 
389 



390 



AUTHOR INDEX 



Bieberbach, L., 256 
Blichfeldt,H.F., 231,256 
Burnside, W., 231, 257, 268, 272 
Dickson, L. E., 253 
Frobenius, G., 256, 257, 274 
Fuchs, L., 209, 215 
Gordan, P., 215 
Jordan, C, 207, 215, 239, 256 
Klein, F., 215, 225, 254 
Kronecker, L., 240 



Loewy, A., 209 
Maschice, H., 207, 253 
Mitchell, H, H., 256 
Molien, T., 257 
Moore, E. H., 209, 250 
Picard, E., 209 
Schur, J., 256, 257 
Valentiner, H., 209, 215, 256 
Wiman, A., 254 



Abel, N, H., 317, 320 

Aronhold, S, 361, 367-71 

Cardan, G., 301 

Cayley, A., 357 

Clebsch, A., 372, 375-7 

Dickson, L. E., 320, 326, 350, 365, 373, 

376-7, 380 
Euler, L., 328 

Galois, E., 298, 315-20, 378 
Gauss, C. F., 309, 326 
Geiser, C. F., 351 
Hermite, C., 378 
Hesse, O., 329, 357 
Holder, O., 318 

Jordan, C, 317, 336, 348, 350, 377-81 
Kempner, A. J., 292 



Klein, F., 377, 378, 381 

Kronecker, L., 310 

Lagrange, J. L., 298, 307 

Maillet, E., 365-6 

Mitchell, H. H., 350 

Picard, E., 380 

Riemann, B,, 372 

Ruffini, P., 317 

Schlafli, L., 366 

Schlesinger, L., 381 

Serret, J. A,, 320 

Steiner, J., 354-77, 375 v 

Vessiot, E., 380 

Weber, H., 320, 336, 366-7, 372, 379 

381 
Zeuthen, H. G., 366 



QA Miller, George Abram 

171 Theory and applications of 

M55 finite groups 1st ed. 



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